Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations

Dissertation zur Erlangung des Doktorgrades

vorgelegt von Johannes Daube an der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg

Februar 2017 Dekan: Prof. Dr. Gregor Herten 1. Gutachter: Prof. Dr. Dietmar Kröner 2. Gutachter: Prof. Dr. Helmut Abels Datum der mündlichen Prüfung: 09.11.2016 Contents

Abstract v

Acknowledgements vii

List of Symbols ix

1 Introduction 1 1.1 PhaseTransitions...... 1 1.2 CapillaryEﬀects ...... 2 1.3 Sharp- and Diﬀuse-Interface Models and the Sharp-Interface Limit . . . . . 3 1.4 The Navier–Stokes–Korteweg Model ...... 4 1.5 TheStaticCase...... 10 1.6 ExistingResults ...... 13 1.7 NewContributions ...... 16 1.8 Outline ...... 16

2 Mathematical Background 19 2.1 Notation...... 19 2.2 Measures ...... 26 2.3 Functions of Bounded Variation ...... 30

3 The Diﬀuse-Interface Model 35 3.1 The Double-Well Potential ...... 35 3.2 TheNotionofWeakSolutions...... 37 3.3 APrioriEstimates ...... 43 3.4 CompactnessofWeakSolutions...... 56 3.5 LimitingInterfaces ...... 60 3.6 Remarks...... 63

4 The Sharp-Interface Model 67 4.1 Two-Phase Incompressible Navier–Stokes Equations with Surface Tension . 67 4.2 Hypersurfaces...... 70 4.3 TheNotionofWeakSolutions...... 73 4.4 Lebesgue and Sobolev Spaces on Time-Dependent Domains ...... 87

iii Contents

4.5 Consistency of the Weak Formulation ...... 94

5 The Sharp-Interface Limit 113 5.1 Assumptions ...... 114 5.2 CapillaryTerm ...... 117 5.3 EnergyInequality...... 121 5.4 Regularity of Limiting Velocity and Transport Equation ...... 124 5.5 Variational Formulation ...... 126 5.6 MainTheorem ...... 127 5.7 Discussion: Convergence of Associated Energy Functional ...... 127

Bibliography 139

iv Abstract

Liquid-vapour flows can be described by two different types of mathematical models: known as diffuse- and sharp-interface models, respectively. The difference between them lies in the representation of the interfacial layer where phase transitions occur. While in diffuse-interface models this region has a small, but positive, thickness, in sharp-interface models an infinitesimally thin hypersurface is used instead. The diffuse-interface model can be related to the associated sharp-interface model by taking the limit where the thickness of the interfacial region tends to zero. We will call this the sharp-interface limit of the diffuse-interface model. Here, we investigate the sharp-interface limit for the Navier–Stokes–Korteweg model, which is an extension of the compressible Navier–Stokes equations. This diffuse-interface model for liquid-vapour flows was already proposed by the Dutch mathematician Diederik Johannes Korteweg in 1901. By means of compactness arguments, we show that solutions of the Navier–Stokes–Korteweg equations converge to solutions of a physically meaningful free-boundary problem. As- suming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions.

The following paper was published prior to the present dissertation: [9].

Acknowledgements

I would like to thank all the people who contributed in some way to the work described in this thesis. First and foremost, I thank my advisor, Prof. Dietmar Kröner, for accepting me into his group and for suggesting the challenging topic that finally resulted in the present work. I would like to express my sincere gratitude to Prof. Christiane Kraus and Prof. Helmut Abels for their guidance and support over the years. They have introduced me to the world of sharp-interface limits. Their valuable insights and constructive comments have greatly improved the quality of this thesis. It was a pleasure to work with them. An additional thanks to Prof. Helmut Abels for providing me with the pictures shown in Section 5.7. I am grateful to Dr Andrew Lam and Dr Philipp Nägele for numerous discussions on related topics that helped me to improve my knowledge in the area. I am also indebted to all the people who read preliminary versions of the present thesis. Their remarks contributed to the final version, clarified the presentation and improved the language. Finally, I must express my very profound gratitude to my parents and to my partner Anna for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them.

vii

List of Symbols

Throughout this work we use the following notation. The numbers following the descrip- tions refer to the page where the symbol is deﬁned.

Real Numbers, Vectors and Matrices

N, N0 natural numbers and non-negative integers 19 R real numbers 19 Rd d-dimensional Euclidean vector space 19 Rk×l vector space of k l matrices 19 × (a, b), [a, b] open and closed interval 19 [a, b), (a, b] half-open intervals 19 Euclidean norm 20 | · | x y scalar product of vectors x and y 19 · x y tensor product of vectors x and y 19 ⊗ A : B scalar product of matrices A and B 19 Bt transpose of a matrix B 20 Ab product of a matrix A and a vector b 19 det(A) determinant of a quadratic matrix A 20

Function Spaces

C0(S) continuous functions 22 Ck(S) k-times continuously differentiable functions 22 Ck([0, )) k-times continuously differentiable functions on [0, ) 22 ∞ ∞ k k C0 (S) compactly supported C -functions 22 ∞ C(0)(Q) restrictions of test functions with support in Q 23 (S) smooth test functions 23 D ′(S) distributions 23 D Ck,1(Ω (0, T )) differentiable function of mixed regularity 23 × 3,1 Cb (Q) bounded, differentiable function of mixed regularity 23 C1(Ω [0, T )) continuously differentiable functions on Ω [0, T ) 23 × ×

ix Topological Notation

∞ C0,σ(Ω) solenoidal test functions 75 ∞ ∞ C0 ((0, T ); C0,σ (Ω)) time-dependent solenoidal test functions 76 ∞ ∞ C0 ([0, T ); C0,σ (Ω)) time-dependent solenoidal test functions 76 r r L (M), Lloc(M) Lebesgue spaces 24 W k,r(D), Hk(D) Sobolev spaces 24 BV (Ω), BV (Ω, M) functions of bounded variation 31 W −1,r(D), H−1(D) Sobolev spaces of negative order 24 1− 1 ,r 1 W r (∂D), H 2 (∂D) Sobolev trace spaces 24, 25 r r L0,σ(Ω) closure of solenoidal test functions in L (Ω) 75 1,r 1 1,r 1 W0,σ (Ω), H0,σ(Ω) closure of solenoidal test functions in W (Ω) and in H (Ω) 75 Lr(0, T ; Y ) Y -valued Bochner space 26 C0([0, T ]; Y ) continuous Y -valued functions 50 C0,α([0, T ]; Y ) H¨older continuous Y -valued functions 50 L∞ (0, T ; Y ∗) Y ∗-valued weakly- measurable functions 116 ω∗ ∗

Diﬀerential Operators

u′ derivative of u 20 ∂ u, ∂u i-th partial derivative of a real-valued function u 20 i ∂xi

∂tu time derivative of a real- or vector-valued function u 21 u gradient of a real- or vector-valued function u 20 ∇ Du symmetric gradient of u 20 div(u) divergence of a vector- or matrix-valued function u 20, 21 (u )u convective term of a vector-valued function u 21 ·∇ ∆u Laplacian of a real- or vector-valued function u 20 π higher-order partial derivative 23 ∇

Topological Notation

int(E), E interior and closure of a set E 20 ∂E topological boundary of a set E 20

Br(x) open ball with radius r and centre x 20 Sd−1 (d 1)-dimensional unit sphere 20 − F E F is compactly contained in E 20 ⊂⊂ dist(x, E) distance of the point x to the set E 20

x Hypersurfaces

Hypersurfaces

TxΓ tangent space of a hypersurface Γ at the point x 70 κ mean curvature 71 [ ] jump across a hypersurface 9, 68, 104 · V normal velocity of an evolving hypersurface 72 C1(Γ) continuously diﬀerentiable functions on a hypersurface Γ 70 , div surface gradient and surface divergence 70 ∇Γ Γ

Measure Theoretic Notation

d, M d-dimensional Lebesgue measure (of a set M) 24 L | | (X) Borel σ-algebra 26 B ess sup essential supremum 24 d−1 (d 1)-dimensional Hausdorﬀ measure 30 H − (X)s, (X) Radon measures 26 M M +(X) non-negative Radon measures 26 M (E, Ω) perimeter of a set E in Ω 32 P µxE restriction of a Radon measure µ to a set E 27 fµ Radon measure induced by a µ-integrable function f 27 µ ν Radon–Nikodym derivative 29 ∗ ∂∗E, ∂ E essential and reduced boundary of a set E 33

Constants

β1, β2 Maxwell points 6 a, b mean value and midpoint of the Maxwell points 35 p∗ constant determining the growth of the double-well potential 35

σst surface-tension constant 12, 62, 63, 69

Miscellaneous

W normalized double-well potential 6, 35

χE characteristic function of a set E 25, 32, 60 Y ∗ dual space of a normed vector space Y 21

xi Miscellaneous

Y ∗ dual norm 21 k · k , Y duality pairing 21 h · · i , ⇀, ⇁∗ strong, weak and weak- convergence 21 → ∗ continuous and compact embedding 22 ֒ ֒ , ֒ → →→

xii Chapter 1

Introduction

1.1 Phase Transitions

In everyday life water occurs in various forms, such as ice, liquid water and water vapour. These different physical states are respectively referred to as the solid, water and vapour (or gas) phases of water. For water vapour one might have in mind the example of boiling water while preparing a cup of tea. However, what we observe coming out of the kettle and filling the kitchen is water steam: a mixture of air, water vapour and fine droplets of water. In the natural sciences, however, water vapour means the (invisible !) gaseous phase of water. Despite this impreciseness in this everyday example, we nevertheless get a rough idea of the vapourisation of water, i.e., a phase transition from the liquid to the vapour phase. Looking at the kitchen window, especially on cold winter days, we see that the water readily transforms back to small droplets of water. This provides an illustration of condensation, the phase transition from the vapour to the liquid phase. In general, between the solid, the liquid and the vapour phase there are six types of phase transitions:

— Melting: solid liquid, −→ — Sublimation: solid vapour, −→ — Freezing: liquid solid, −→ — Vapourisation: liquid vapour, −→ — Deposition or desublimation: vapour solid, −→ — Condensation: vapour liquid. −→ In the example of preparing tea, the phase transition is driven by a change of temperature: heating up the water leads to vapourisation and as the cold surface of the window cools down, the vapour steam condensates. We observe the two phenomena of phase change at constant pressure. In general, the question as to which phase matter exists in depends not only on the eﬀects of temperature, but also on the eﬀects of pressure. This dependence on the pressure can be observed in the mountains, where water boils at a lower temperature than at the seaside. This is due to the lower pressure at high altitude. Figure 1.1, which

1 Chapter 1 Introduction is taken from [93], shows the water cycle of the earth and illustrates the phase transitions of water.

Figure 1.1: Schematic illustration of the global water cycle.

In contrast to solids, liquids and gases both have the ability to flow. Together they form the class of the fluids. However, they significantly differ in their mass densities. Assuming constant temperature, this allows us to use the mass density to distinguish different phases. In this thesis we shall consider fluids at constant temperature and we will focus on phase transitions between liquids and gases, i.e., vapourisation and condensation. When considering a container filled with a fluid, the phase boundaries separating the liquid from the vapour phase are of natural interest. Due to the difference in the mass densities, we expect phase boundaries to be regions where the density function has steep gradients or even jumps, i.e., is discontinuous. This gives rise to two different ways of representing phase boundaries: either as thin regions of steep density gradients or as infinitesimally thin regions of density jumps. These kinds of models are referred to as diffuse-interface models and sharp-interface models, respectively.

1.2 Capillary Eﬀects

In everyday life we are very familiar with capillary effects: for example, tissue absorbs liquid from a surface, a raindrop forms, the ink inside a pen is transported to its tip and a candle burns. In the last example, the candle’s wick draws the melted wax up to the flame by capillary action. There, once the wax reaches the flame, it vapourises and combusts.

2 1.3 Sharp- and Diﬀuse-Interface Models and the Sharp-Interface Limit

Returning to the tea example, one sees the same effect by putting a thin tube into a cup of tea: the tea enters the tube and rises up the inside. At the top, the surface of the tea forms a concave meniscus. Whether the surface is concave or convex depends on the material; mercury, for example, forms convex menisci (an effect known from mercury-in- glass thermometers [21]). This phenomenon of liquids in thin tubes was already observed in medieval times (or even earlier). Due to the lack of an explanation at that time, it was described by the Latin word capillus, meaning hair [42]. Nowadays, we know that this effect is caused by the surface tension of the tea itself and the interfacial tension between the liquid and the solid surface of the tube. The development of the modern theory of capillarity dates back to the 19th century. Rely- ing heavily on mathematical methods, especially calculus, calculus of variations and geometry, Young [100] and Laplace [64] independently established the famous Young–Laplace relation. Later in 1830, Gauß [46] unified the work of Young and Laplace. They considered the interface between two fluids at rest, represented as a surface of zero thickness, and related the difference (jump) in pressure of the fluids across the interface to the product of its surface tension and its curvature:

[p] = 2σstκ, (1.2.1) where [p] denotes the pressure jump, σst the surface-tension constant and κ is the mean curvature, i.e., the sum of the principal curvatures of the interface. From the mathematical point of view, (1.2.1) is a boundary condition at the interface. Nowadays, the identity (1.2.1) is known as the Young–Laplace law. In contrast to the ideas of Young, Laplace and Gauß; Poisson [79], Maxwell [72] and Gibbs [47] came up with the observation that the interface separating the two ﬂuids is actually a thin transition layer, where physical quantities undergo a rapid but smooth change. This led to the idea of considering diﬀuse interfaces of non-zero thickness. From thermodynamic principles, Lord Rayleigh [68] and van der Waals [96] derived gradient theories for the interface. Based on these ideas, Korteweg [60], a student of van der Waals, proposed in 1901 a constitutive law for the capillary stress depending not only on the density, but also on its gradient. For a more complete overview and additional references on capillary and interfacial phenomena we refer to the aforementioned references [21, 42] and additionally to [17, 85].

1.3 Sharp- and Diﬀuse-Interface Models and the Sharp-Interface Limit

Models describing liquid-vapour flow for compressible fluids are basically classified into two different types: sharp- and diffuse-interface models. They differ in how the interface dividing liquid from vapour is represented.

3 Chapter 1 Introduction

In sharp-interface models, the interface is represented as a hypersurface. Physical quantities, such as density or pressure, allow for (jump) discontinuities across the interface. Other quantities, like surface tension may only be defined at the interface. From a mathematical point of view, sharp-interface models are free-boundary problems, since the position of the interface is a priori unknown and thus is part of the solution to the free-boundary problem. In the liquid and vapour phase, respectively, a distinct system of partial differential equations describes the motion of the fluid. These systems are coupled by boundary conditions at the interface. In diffuse-interface models, an additional variable, the “order parameter“1 is introduced, such that the interface is described in a different manner. The “sharp interface“ is replaced by an interfacial layer of positive thickness ε. In this region, the order parameter varies rapidly but smoothly between two values distinguishing the liquid and the vapour phase. The behaviour of the order parameter is described by an additional partial differential equation. The motion of the fluid is governed by a single system of partial differential equations on the whole domain, incorporating the thickness parameter ε. Despite the fact that sharp-interface models are quite intuitive from the physical point of view, they possess a few drawbacks. In numerical simulations, one has to track the interface explicitly, as its position is a priori unknown. Often, this turns out to be a difficult task. Moreover, sharp-interface models are not appropriate for describing topology changes of interfaces (e.g. merging and pinching), as they require certain regularity assumptions. Classical numerical methods fail to describe these types of phenomena. Diffuse-interface models provide an approach for overcoming these difficulties; since interface tracking is no longer necessary, one may hope for the numerical treatment to become easier. Considering an interfacial layer, instead of a smooth hypersurface, leads to a better treatment of changes of the interface topology and it is natural to investigate the limiting relation between sharp- and diffuse-interface models. The sharp-interface limit ad- dresses the behaviour of diffuse-interface models and their corresponding solutions, as the interface thickness ε tends to zero. Hence, sharp-interface models are obtained as limits of diffuse-interface models. Alternatively, a diffuse-interface model can be regarded as an approximation or regularization of a sharp-interface model.

1.4 The Navier–Stokes–Korteweg Model

Already in 1901, Korteweg [60] proposed a mathematical model for liquid-vapour ﬂows including phase transitions; see also [96]. This model, known as the Navier–Stokes–Korteweg system, is an extension of the compressible Navier–Stokes equations. In the isothermal

1In the case of the Navier–Stokes–Korteweg equations, treated here, the density is used as the order parameter.

4 1.4 The Navier–Stokes–Korteweg Model case the ﬂuid’s density ρ and its velocity v satisfy the following system of partial diﬀerential equations:

∂tρ + div(ρv) = 0, (1.4.1) ∂t(ρv) + div(ρv v)+ p(ρ) = div(τ + K), ⊗ ∇ in the space-time cylinder Ω (0, T ), with Ω Rn open and bounded, where τ and K × ⊂ denote the viscous and the Korteweg stress tensor, respectively. They are given explicitly by τ = τ(ρ, Dv) = 2µ(ρ)Dv, where µ = µ(ρ) is the density-dependent viscosity of the ﬂuid, Dv = 1 v + ( v)t 2 ∇ ∇ denotes the symmetric part of the gradient, and

K = K(ρ, ρ)= γ ρ∆ρI + 1 ρ 2 I ρ ρ , ∇ 2 |∇ | −∇ ⊗∇ where γ > 0 is the capillary constant and I Rn×n denotes the identity tensor. Moreover, ∈ p = p(ρ) denotes the pressure of the ﬂuid. For a rigorous derivation of the system (1.4.1) we refer to [18, 35, 89]. Compared to the compressible Navier–Stokes equations, the system (1.4.1) contains the additional term

div(K) = div(K(ρ, ρ)) = γρ ∆ρ. ∇ ∇ The idea of introducing a stress tensor depending not only on the density ρ, but also on its gradient ρ, in order to model capillary eﬀects close to phase transitions goes back to ∇ van der Waals [96].

1.4.1 Non-Monotone Pressure Law and Double-Well Potential

The pressure function p = p(ρ) in (1.4.1) is given by the van der Waals equation of state

p(ρ)= ρ2ψ′(ρ), (1.4.2) for a given (smooth) specific Helmholtz energy ψ = ψ(ρ), which, in the isothermal case, depends only on the density. In order to model phase transitions, we assume that the Helmholtz free energy W (ρ)= ρψ(ρ) has a double-well shape. The functions p and W are related by f f p(ρ)= ρW ′(ρ) W (ρ) (1.4.3) − and therefore, due to the double-well shapef of W , thef pressure is a non-monotone function of the density. Due to its non-monotone nature, we expect the pressure function to have f two local extreme values at some points α1 < α2. The liquid and the vapour phase of the fluid are now characterised by the density: if ρ α the fluid appears as vapour, ≤ 1 whereas ρ α indicates the liquid phase. For values ρ (α , α ), we are in the so-called ≥ 2 ∈ 1 2

5 Chapter 1 Introduction

Figure 1.2: Typical shape of p. spinodal or elliptic region, where liquid and vapour coexist. This situation is illustrated in Figure 1.2. Moreover, there exist the Maxwell points β1 < β2, see Figure 1.3, at which the tangent line of W has slope equal to the Newton quotient, i.e.,

f ′ ′ W (β2) W (β1) W (β1)= W (β2)= − , (1.4.4) β2 β1 f − f f f and there holds W ′′(β ), W ′′(β ) > 0. The Maxwell line l(ρ) = W ′(β )(ρ β )+ W (β ) 1 2 1 − 1 1 then satisﬁes l(β )= W (β ) and l′(β )= W ′(β ) for i = 1, 2. if if i i f f f f

Figure 1.3: Maxwell construction.

Note that (1.4.3) and (1.4.4) imply that p(β1) = p(β2). Throughout this thesis we will always consider the normalized double-well potential

W (ρ)= W (ρ) l(ρ)= ρψ(ρ) l(ρ) (1.4.5) − − f instead of W . Since, by the deﬁnition of W , subtracting the Maxwell line does not change second derivatives, we have W ′′ = W ′′ and, in view of (1.4.3), there holds f f p′(ρ)= ρW ′′(ρ)= ρW ′′(ρ), (1.4.6)

6 1.4 The Navier–Stokes–Korteweg Model

As only p′(ρ) appears in (1.4.1), this allow us to replace W by W . Subtracting the Maxwell line from W ensures that W has two zeros at β and β which are both local minimal 1 f2 values of W . This situation is exempliﬁed in Figure 1.4. f

Figure 1.4: Typical shape of W .

Precise mathematical assumptions on W will be formulated in Section 3.1.

1.4.2 Phase-Field-Like Scaling

To identify the physically signiﬁcant quantities of (1.4.1) and to understand the physical meaning of its diﬀerent scalings, we have to non-dimensionalise the system. For this ¯ x¯ purpose, we introduce reference quantitiesx ¯, t,p ¯,ρ ¯,v ¯ = t¯,µ ¯ and denote the speed p¯ of sound byc ¯ = ρ¯. The parameter ε > 0 represents the thickness of the interface. Considering recoveryq sequences of the Γ-limit of the functional (1.5.1) motivates this choice of scaling; see [76, 92] and cf. also [9, Remark 1.1]. The non-dimensionalised form of the Navier–Stokes–Korteweg equations [51, equation (18)], now in Ω (0, T ), reads ×

∂tρ + div(ρv) = 0, (1.4.7) 1 2 λ ∂t(ρv) + div(ρv v)+ 2 p(ρ)= div(µ(ρ)Dv)+ 2 ρ ∆ρ, ⊗ M ∇ Re M ∇ where v¯ ρ¯v¯x¯ ρ¯2γε2 M= , Re = , λ = , c¯ µ¯ x¯2p¯ denote the Mach, the Reynolds and the capillarity number, respectively. Hermsd¨orfer, Kraus and Kr¨oner [51] studied the low-Mach-number limit of the Navier– Stokes–Korteweg equations in the static case, which we will consider more thoroughly in Section 1.5. Their results suggest a relation of the interface thickness ε to the Mach number M by ε = M2 as well as the choice λ = M4 and Re = 1. Using this choice in (1.4.7) results in the system

∂tρε + div(ρεvε) = 0, (1.4.8) 1 ∂t(ρεvε) + div(ρεvε vε)+ p(ρε) = 2div(µ(ρε)Dvε)+ ερε ∆ρε ⊗ ε ∇ ∇

7 Chapter 1 Introduction

in Ω (0, T ). In accordance with [51], we will call the system (1.4.8) “phase-field-like × scaling” of the Navier–Stokes–Korteweg equations in the sequel. The results of Hermsdörfer, Kraus and Kröner serve as staring point for this thesis: we will investigate the sharp-interface limit of the phase-field-like scaling in the dynamic case.

We are interested in the behaviour of (weak) solutions (ρε, vε) to (1.4.8) as the parameter ε tends to zero. We seek to discover suitable compactness properties of the sequence

(ρε, vε)ε>0 which allow us to extract subsequences converging to limit functions (ρ0, v0) as ε 0 in an appropriate manner. Moreover, for (ρ , v ) we aim to discover a physically → 0 0 meaningful model (the so-called sharp-interface model). In particular, in accordance with the results of Hermsd¨orfer, Kraus and Kr¨oner, we shall derive a sharp-interface model incorporating a dynamic version of the Young–Laplace law (1.2.1).

1.4.3 The Sharp-Interface Limit

The study of the sharp-interface limit for the phase-ﬁeld-like scaling of the Navier–Stokes– Korteweg equations (1.4.8) is basically divided into two steps. Firstly, we explore the

compactness properties of solutions (ρε, vε) to (1.4.8). Secondly, we show that limit func-

tions (ρ0, v0) of appropriate subsequences of (ρε, vε) are solutions of a physically meaningful sharp-interface model.

Energy considerations imply that (ρε, vε) converge as ε tends to zero, at least along appro-

priate subsequences, to limit functions (ρ0, v0). Classical arguments due to Modica and

Mortola [73, 74] ensure that ρ0 takes only the values β1 and β2 and thus leads to a ”sharp“ limiting interface. Indeed, ρ ( ,t) induces a partition Ω = Ω−(t) Ω+(t) of Ω via 0 · ∪ Ω−(t)= x Ω : ρ (x,t)= β and Ω+(t)= x Ω : ρ (x,t)= β . { ∈ 0 1} { ∈ 0 2} A slight modiﬁcation of this partition, namely

Ω−(t) = int x Ω : ρ (x,t)= β and Ω+(t)=Ω Ω−(t), { ∈ 0 1} \ gives rise to the (pairwise disjoint) partition Ω = Ω−(t) Γ(t) Ω+(t), where ∪ ∪ Γ(t)= ∂Ω−(t) Ω ∩ is the interface separating Ω into sets Ω−(t) and Ω+(t), which are open, thanks to the modiﬁcation. Let us assume that in (1.4.8) we may pass to the limit ε 0 in the conservation-of-mass → equation ∂tρε + div(ρεvε) = 0, i.e., that

∂tρ + div(ρ v )=0inΩ (0, T ) 0 0 0 × − + holds. This, in turn, as ρ0 is constant in Ω (t) and Ω (t), respectively, leads to

− β div(v ( ,t)) = ∂tρ ( ,t) + div(ρ ( ,t)v ( ,t))=0 in Ω (t) 1 0 · 0 · 0 · 0 ·

8 1.4 The Navier–Stokes–Korteweg Model and + β div(v ( ,t)) = ∂tρ ( ,t) + div(ρ ( ,t)v ( ,t))=0 in Ω (t). 2 0 · 0 · 0 · 0 · Hence, v ( ,t) satisﬁes the incompressibility condition 0 · div(v ( ,t))=0 in Ω Γ(t). 0 · \ In this thesis we shall not only make the above heuristic ideas rigorous by establishing appropriate compactness properties of the sequence (ρε, vε)ε>0, but we will also prove convergence of (ρε, vε)ε>0 to solutions of the sharp-interface model below, provided the convergence of an associated in a suitable sense.

1.4.4 The Sharp-Interface Model

The sharp-interface model is a free-boundary problem describing the motion of the vapour phase, of constant density β1, and the liquid phase, of constant density β2, of an isothermal, viscous, incompressible Newtonian fluid. The two phases are separated by an unknown, infinitesimally thin interface. The motion of the vapour and the liquid phase are governed by the two-phase Navier–Stokes equations with surface tension [59]. For t [0, T ], the ∈ unknowns are the free boundary Γ(t), the velocity field v( ,t):Ω Γ(t) Rn and the · \ → pressure function p( ,t):Ω Γ(t) R. The sharp-interface model then reads as · \ → − β ∂tv + β (v )v µ(β )∆v + p =0 inΩ (t), (1.4.9) 1 1 ·∇ − 1 ∇ + β ∂tv + β (v )v µ(β )∆v + p =0 inΩ (t), (1.4.10) 2 2 ·∇ − 2 ∇ div(v)=0 inΩ Γ(t), (1.4.11) \ [v]=0 onΓ(t), (1.4.12) V = v ν− on Γ(t), (1.4.13) · [T ] ν− = 2σ κν− on Γ(t) (1.4.14) − st for t [0, T ]. Here, the stress tensor T = T (v,p) is given by ∈

− 2µ(β1)Dv(t) p(t)I in Ω (t), T (v(t),p(t)) =  − 2µ(β )Dv(t) p(t)I in Ω+(t).  2 − For a given quantity f, we denote the jump across the interface Γ(t) in the direction of the exterior unit-normal ﬁeld ν−( ,t) of ∂Ω−(t) by · [f](x,t) = lim f(x + ξν−(x,t),t) f(x ξν−(x,t),t) for x Γ(t). ξց0 − − ∈ As before, the constants β1 and β2 are the Maxwell points and µ(β1) and µ(β2) denote the viscosity of the vapour and the liquid, respectively. Moreover, V and κ are the normal − velocity and the mean curvature of Γ, both taken with respect to ν , and σst is the surface-tension constant.

9 Chapter 1 Introduction

1.5 The Static Case

The limiting behaviour of minimizers of the energy functional

1 2 2 W (ρε)+ ε ρε dx 2 |∇ | ZΩ under the prescribed mass constraint has been investigated by Luckhaus and Modica [69, 73] and Dreyer and Kraus [38]. The results of Dreyer and Kraus lead to a vanishing pressure jump in the limit ε 0. In order to fix this inconsistency with classical hydrodynamics, → Hermsdörfer, Kraus and Kröner [51] propose rescaling the energy functional by a suitable power of ε, more precisely by 1 . In this way, for ε 0, the energy ε → 1 1 2 2 Eε(ρε)= W (ρε)+ ε ρε dx (1.5.1) ε 2 |∇ | ZΩ converges to some limit other than 0 or as ε 0 and for a corresponding rescaled pres- ∞ → ′ 1 ′ sure function pε, with pε = ε p , they recover the Young–Laplace law in the limit, i.e., the scaling (1.5.1) leads to a sharp-interface model including surface energy and incorporating a non-vanishing jump condition for the pressure across the interface. These results motivate the study of the sharp-interface limit of the phase-field-like scaling in the dynamic case. For a better understanding it is convenient to emphasize the relation of the energy functional Eε to the phase-field-like scaling (1.4.8) in the static case; that is, we consider the time-independent situation and assume that the velocity vε vanishes on the whole domain Ω. Then (1.4.8) reduces to 1 p(ρε)= ερε ∆ρε in Ω. ε∇ ∇ In view of (1.4.6), there holds p′(ρ)= ρW ′′(ρ) for any ρ R. Therefore, we have ∈ 1 ′ 1 ′ ερε ∆ρε = p (ρε) ρε = ρε W (ρε) in Ω. ∇ ε ∇ ε ∇

Assuming that ρε > 0, we infer that

1 ′ W (ρε) ε∆ρε =0inΩ. ∇ ε −

Hence, there exists a constant lε R such that ∈ 1 ′ W (ρε) ε∆ρε = lε in Ω. (1.5.2) ε − We may observe that (1.5.2) is the Euler–Lagrange equation, with corresponding Lagrange multiplier lε, of the following minimization problem:

1 min Eε(ρ) : ρ H (Ω), ρ(x)dx = m , (1.5.3) ∈ ZΩ 1 where the functional Eε : H (Ω) [0, ) is deﬁned by (1.5.1) and m (β Ω , β Ω ) is → ∞ ∈ 1 | | 2 | | the prescribed total mass.

10 1.5 The Static Case

For a heuristic picture of the mechanism of minimization (for ﬁxed ε> 0), one can consider both contributions to Eε separately: On the one hand, the double-well term

1 W (ρ)dx ε ZΩ favours those conﬁgurations taking values close to the ”pure phases“ β1 and β2, due to the 1 double-well shape of W . Therefore, minimizing Ω ε W (ρ)dx leads to phase separation. On the other hand, the gradient term R

1 ε ρ 2 dx 2 |∇ | ZΩ penalizes steep gradients, i.e., rapid transitions between the subregions of Ω, where ρ approximately equals β1 and those where ρ is approximately equal to β2. Having these eﬀects in mind, we expect solutions ρε to the minimization problem (1.5.3) in most part of the domain to be close to either of the values β1 or β2 with the corresponding regions divided by an interface with a thickness which is proportional to the parameter ε.

1.5.1 Minimizers and Γ-Convergence

The appropriate abstract framework for studying the asymptotic behaviour of minimisa- tion problems is the concept of Γ-convergence; see for example [26, 71] for further details.

Deﬁnition 1.5.1 (Γ-convergence). Let (X, dX ) be a metric space and (Fε)ε>0 a family of functionals Fε : X [ , ]. We say that (Fε)ε> Γ-converges to a functional F : X → −∞ ∞ 0 → [ , ] as ε 0 if the following conditions hold. −∞ ∞ → 2 1. Lower bound inequality. For every u X and every sequence (uε)ε> X such ∈ 0 ⊂ that uε u in X for ε 0, there holds F (u) lim infε Fε(uε). → → ≤ →0

2. Upper bound inequality. For every u X, there exists a sequence (uε)ε> X such ∈ 0 ⊂ that uε u in X for ε 0 and lim sup Fε(uε) F (u). The sequence (uε)ε> is → → ε→0 ≤ 0 called a recovery sequence of u.

1 The functional Eε extends to L (Ω) by setting

1 1 2 1 Ω ε W (ρ)+ 2 ε ρ dx, for ρ H (Ω), Ω ρ(x)dx = m, Eε(ρ)=  |∇ | ∈ (1.5.4) R , elsewhere in L1(Ω)R . ∞  The L1(Ω)-topology provides the appropriate notion of convergence. For ε 0, the → 1 functional Eε Γ-converges to a multiple of the perimeter functional E : L (Ω) [0, ], → ∞ 2 To be precise, we consider an uncountable inﬁnite family indexed by the positive real numbers, which is not a sequence in the proper sense. With abuse of language, here and in the sequel however, we will use the term “sequence” in order to indicate that it is ordered.

11 Chapter 1 Introduction

given by

2σst χ(ρ) (Ω), for ρ BV (Ω, β1, β2 ), Ω ρ(x)dx = m, E(ρ)=  |∇ | ∈ { } (1.5.5)  , elsewhere in L1(Ω), R ∞  where the constant σst is given by

β 2 W (z) σst = 2 dz (1.5.6) Zβ1 q and χ is deﬁned by χ(ρ) = ρ−β2 . Due to results of Sternberg [92], cf. also [73, 74], we β1−β2 have the following result.

Theorem 1.5.2 (Γ-convergence of Eε). Let the family (Eε)ε>0 be given by (1.5.4). Then, 1 as ε 0, Eε Γ-converges in L (Ω) to the functional E deﬁned by (1.5.5). → Proof. See [92, Section 1 B.].

The following compactness property of minimizers of the functional Eε justiﬁes the choice of the L1(Ω)-topology. It can be achieved if the double-well potential satisﬁes a suitable growth condition; see for example [92, equation (1.32)].

Theorem 1.5.3 (Compactness of minimizers of Eε). For a sequence (εj )j∈N of positive numbers with limj→∞ εj = 0, let (ρεj )j∈N be a sequence of associated minimizers of Eεj given by (1.5.4). If for all j N, there holds Eε (ρε ) C, for some constant C > 0, then ∈ j j ≤ 1 the sequence (ρεj )j∈N is precompact in L (Ω).

Proof. The proof can be found in [92, Proposition 3] if the double-well potential obeys an appropriate growth condition.

As an important consequence, we obtain, by combining the Γ-convergence and the compactness property of minimizing sequences, the convergence of minimizers.

Corollary 1.5.4 (Stability of minimizers). Let Eε and E be as in (1.5.4) and (1.5.5). 1 Furthermore, let (ρε)ε> L (Ω) be a sequence of associated minimizers of Eε. 0 ⊂

1. If there exists some constant C > 0 such that Eε(ρε) C for all ε> 0, then there exist ≤ 1 1 a subsequence (ρε )j N and a limit function ρ L (Ω) such that ρε ρ in L (Ω) j ∈ 0 ∈ j → 0 for j and ρ minimizes the functional E. →∞ 0

1 2. Every cluster point (with respect to L (Ω)-convergence) of (ρε)ε>0 is a minimizer of E.

Proof. In view of Theorem 1.5.3, without loss of generality, we may assume that ρε ρ → 0 in L1(Ω) as ε 0 for some ρ L1(Ω), and it is suﬃcient to show that ρ minimizes E. To → 0 ∈ 0 this end, we arbitrarily choose some function ρ L1(Ω). Then, in view of Theorem 1.5.2 ∈

12 1.6 Existing Results and Deﬁnition 1.5.1, there exists a recovery sequence (ρ∗) L1(Ω) for ρ with ρ∗ ρ in ε ⊂ ε → L1(Ω) as ε 0 and → ∗ lim sup Eε(ρε) E(ρ). ε→0 ≤

We then, by Deﬁnition 1.5.1 and the fact that ρε minimizes Eε, conclude that

∗ E(ρ0) lim inf Eε(ρε) lim sup Eε(ρε) E(ρ). ≤ ε→0 ≤ ε→0 ≤

Hence ρ0 is a minimizer of the functional E.

Besides its importance for the Γ-convergence of the functional Eε, the validity of The- orem 1.5.3 is not restricted to the case of minimizers to Eε. Its proof is based on ideas going back to Modica and Mortola [73, 74]. These ideas have been generalised and adapted to many situations; see for example [6, 29, 54, 65, 69]. In the course of this work we shall adapt the underlying ideas for our own purposes. The growth condition for the double-well potential will be formulated precisely in (3.1.1); the arguments leading to a statement in the spirit of Theorem 1.5.3 will be carried out in Section 3.3.2.

1.5.2 Lagrange Multipliers

For the problems (1.5.2) and (1.5.3), the asymptotic behaviour as ε 0 is well-studied. → The Euler–Lagrange equation (1.5.2) is the content of [69]. In scope of this work, Luckhaus and Modica prove (up to the choice of a subsequence) the following asymptotic behaviour 1 as ε tends to zero. In the L (Ω)-sense there holds ρε ρ as ε 0, where the limit → 0 → function ρ0 takes only the values β1 and β2, and the interface

Γ= ∂Ω− Ω, where Ω− = x Ω : ρ (x)= β , ∩ { ∈ 0 1} is smooth and has constant mean curvature κ. The Lagrange multiplier lε satisﬁes

2σstκ lim lε = , ε→0 −β β 2 − 1 where σst is given by (1.5.6). This result is consistent with the Γ-convergence of Eε; see Theorem 1.5.2. In the same spirit are the results of Hutchinson and Tonegawa [54], who studied the asymptotic behaviour of critical points of the functional Eε for ε 0 in a → varifold context.

1.6 Existing Results

We now give an overview, which is by no means exhaustive, on the literature related to the topics of this thesis. Additional information and further references can be found in the survey articles [4, 17, 44] and reference therein.

13 Chapter 1 Introduction

1.6.1 Navier–Stokes–Korteweg Equations

The question of well-posedness of the Navier–Stokes–Korteweg system (1.4.1) and related models is the content of several works. Nevertheless, we have to stress that, to the best of our knowledge, there are no global-existence results for solutions of the phase-ﬁeld-

like scaling available that ensure the existence of a suitable family of solutions (ρε, vε)ε>0 to (1.4.8), cf. Definition 3.2.5 below and see also the discussion in Remark 3.2.7. The non-monotonicity of the pressure function is a major source of difficulty in the analytic treatment, whereas it is a crucial requirement in modelling two-phase flow.

For monotone pressure functions, the well-posedness of (1.4.1) has been established in several diﬀerent contexts. Hattori and Li [50] proved the global existence of smooth solutions in two spatial dimensions, provided the initial data satisfy a smallness condition. Bresch, Desjardins and Lin [27] ensured well-posedness in the case of periodic boundary conditions. In the framework of weak solutions, the proof of well-posedness of (1.4.1) is due to Haspot [49], cf. also Remark 3.2.7. Danchin and Desjardins [30] studied the well- posedness of (1.4.1) on the whole space Rd, d 2, for non-monotone pressure functions ≥ of van der Waals type. They are able to obtain well-posedness results if either the initial data are close enough to state equilibria or the initial densities are bounded away from zero, i.e., one is suﬃciently far from vacuum. The non-dissipative case of (1.4.1), the so-called Euler–Korteweg equations, has been investigated by Benzoni-Gavage, Danchin and Descombes. In [22, 23] they prove well-posedness of the Cauchy problem for the Euler–Korteweg equations.

For non-monotone pressure functions, Kotschote [61, 62] proved short-time existence of strong solutions to (1.4.1); see also Remark 3.2.7.

1.6.2 Two-Phase Incompressible Navier–Stokes Equations

The question of (unique) solvability of (1.4.9)–(1.4.14) and related systems has been studied by many authors. In the framework of Hölder spaces, Denisova and Solonnikov first studied the corresponding two-phase Stokes problem [33]. Later they proved well- posedness of (1.4.9)–(1.4.14) for appropriate initial data [34]. Existence results for (1.4.9)– (1.4.14) in the context of maximal Lq-regularity are due to Prüss and Simonett [82] and Köhne, Prüss and Wilke [59]. In view of the interface condition (1.4.13), the interface is purely transported by the fluid and no phase transition occurs. The case of phase transitions has been investigated, also in the framework of maximal Lq-regularity, by Prüss and Shimizu [80] and together with Wilke [81].

14 1.6 Existing Results

1.6.3 Sharp-Interface Limits and Asymptotic Behaviour

1.6.3.1 Formal Asymptotic Expansions

A frequently used technique for obtaining the sharp-interface limit of a phase-field model is the method of formally matched asymptotic expansions. For general information about this formal method of deriving sharp-interface models we refer to Section 5.7.2.2 and to the works cited below and references given there. The sharp-interface limit of the Navier–Stokes–Korteweg equations is studied by Dreyer, Giesselmann, Kraus and Rohde [37]. In contrast to the present thesis, these authors investigated scalings which are different to the phase-field-like scaling (1.4.8). As a further difference, the authors suppose in [37] that, in the limit, the transport condition (1.4.13) is violated. Due to this assumption, in the (formal) limit a sharp-interface model is obtained that incorporates phase-transition effects. A diffuse-interface model for two incompressible, viscous Newtonian fluids of different densities is proposed by Abels, Garcke and Grün. The model is of Navier–Stokes/Cahn– Hilliard type and a generalisation of the so-called ”Model H“ due to Hohenberg and Halperin [53]. In [5] the sharp-interface limit for different scalings of the mobility for the former model is obtained. The limiting behaviour of the compressible Navier–Stokes equations coupled to an Allen–Cahn equation is studied by Alt and Witterstein [14, 98]. A quasi-incompressible phase-field model, i.e., one which comprises a mixture of two incompressible constituents, such that the total mixture is not incompressible, is investigated by Aki and co-authors. In [9, 10] they study the sharp-interface limit of a Navier–Stokes– Korteweg/Cahn–Hilliard/Allen–Cahn-type model.

1.6.3.2 Rigorous Convergence Results

The works cited above show that there are a couple of sharp-interface-limit results on the formal level which can be obtained with the technique of formally matched asymptotic expansions. In contrast, mathematically rigorous convergence results are rare. Ilmanen [55] and Chen [29] performed the sharp-interface limit, respectively, for the Allen–Cahn and the Cahn–Hilliard equation in a varifold framework. The ideas of Chen have been generalised to related problems, for example, by Garcke and Kwak [45] to a Cahn–Larchésystem and by Abels and Lengeler [6] to the aforementioned model due to Abels, Gracke and Grün. In Chapter 3 we will adapt the ideas of Chen to the Navier–Stokes–Korteweg model. Fur- ther methods for performing sharp-interface limits for the Cahn–Hilliard equation are due to Alikakos, Bates and Chen [11] and Le [65]. A generalisation of the techniques of the former work can be found in [88]. In the recent literature there are several convergence results for different scalings of the Navier–Stokes–Korteweg equations, where the corresponding sharp-interface limits are

15 Chapter 1 Introduction performed in the case of monotone pressure functions. The zero-Mach-number limit on the two-dimensional torus is studied by J¨ungel, Lin and Wu [57] and on the three-dimensional torus, as well as on the whole space R3, by Li and Yong [66]. The vanishing-capillarity limit on the three-dimensional whole space is investigated by Bian, Yao and Zhu [24]. In contrast to the aforementioned results, in this thesis, we study the sharp-interface limit for non-monotone pressure functions. This leads (under appropriate assumptions) to a two-phase free-boundary problem in the limit, which does not occur for monotone pressure functions.

1.7 New Contributions

In this thesis we present the first rigorous investigation of the sharp-interface limit for the Navier–Stokes–Korteweg equations at the level of weak solutions in the case of a non- monotone pressure function. Our proceeding is based on the convergence of an associated energy functional. In view of our results, this additional condition (see Assumptions 5.1.6 and the discussion in Section 5.7, and Assumptions 5.3.1) is sufficient to show convergence to a physically meaningful sharp-interface model incorporating the Young–Laplace law. Our work extends that of Hermsdörfer, Kraus and Kröner [51], where the static case of the Navier–Stokes–Korteweg equations is treated, to the dynamic case and closes a mathematical gap in [63]. Earlier results in this direction by Dreyer, Giesselmann, Kraus and Rohde [37] were merely performed at the level of formally matched asymptotic expansions. Moreover, these results are only valid for scalings other than the phase-field-like scaling of the Navier–Stokes–Korteweg equations. The studied scalings do not lead to sharp-interface models incorporating the Young–Laplace law. In earlier results [24, 57, 66], the sharp-interface limit of the Navier–Stokes–Korteweg equations was obtained for monotone pressure functions. It is worth pointing out that this, different to the situation here, does not lead to two-phase free-boundary problems in the limit.

1.8 Outline

The investigation of the sharp-interface limit for the Navier–Stokes–Korteweg equations naturally divides into three main parts, namely the study of

Part 1: the diﬀuse-interface model (the Navier–Stokes–Korteweg equations),

Part 2: the sharp-interface model and

Part 3: the sharp-interface limit relating the diﬀuse and the sharp-interface model.

The organisation of this thesis is as follows:

16 1.8 Outline

We start with some preparatory material in Chapter 2. After introducing the basic notation, we summarise some mathematical background material that we will use throughout this thesis. In Chapter 3 we study the diffuse-interface model. Based on energy considerations, we derive a weak formulation of the Navier–Stokes–Korteweg equations. We then prove a priori estimates for weak solutions that are uniform in the parameter ε which is related to the interface thickness. These estimates serve as the starting point for performing the sharp-interface limit. They allow compactness arguments to be used and ensure existence of converging subsequences in suitable function spaces as ε tends to zero. For the proof of these compactness results, we basically rely on classical ideas going back to Modica and Mortola [73, 74], which have been successfully applied to similar problems by many authors. To cite only a few we mention [29, 54, 65, 69, 94, 95]. We adapt these ideas to the Navier–Stokes–Korteweg equations and derive suitable a priori estimates with respect to the spatial variables. In order to derive a priori estimates which allow for control with respect to the time variable, we use techniques due to Chen [29]. Originally these ideas were applied to perform the sharp-interface limit for the Cahn–Hilliard equation. In this thesis we present these ideas and adapt them in such a way that they can be applied to the Navier–Stokes–Korteweg equations. In Chapter 4 we present the sharp-interface model. Throughout this thesis we will refer to this model as the two-phase incompressible Navier–Stokes equations with surface tension. We take this name from [82]. In the sense of Abels [2] we shall derive an appropriate weak formulation. However, for our purposes we need to generalise the ideas of Abels to the case of two phases of different densities β β , whereas in [2] the equal-density case 1 ≤ 2 β1 = β2 = 1 is explored. Much effort is then needed to justify this notion. We provide the functional-analytic background for the treatment of functions on time-dependent domains. To this end, we generalise the usual Bochner spaces in an appropriate manner. Similar approaches have been successfully applied e.g. in [12, 13, 75, 87]. We conclude this chapter with a, to the best of our knowledge, new consistency result which justifies the notion of weak solutions. In Chapter 5 we link the results concerning the diffuse- and the sharp-interface model. We prove our main result which is, as far as we know, new and perform the sharp-interface limit for the Navier–Stokes–Korteweg model, i.e., we show that weak solutions of the diffuse-interface model converge to weak solutions of the sharp-interface model, assuming the convergence of an associated energy functional.

Chapter 2

Mathematical Background

In this preparatory chapter we shall provide mathematical background material. We will start with introducing the basic notation in Section 2.1. In the remainder of this chapter we will recall some basic deﬁnitions and facts about measures in Section 2.2 and about functions of bounded variation in Section 2.3. To simplify the presentation, we will present additional mathematical background material in the particular situation when it is needed.

2.1 Notation

We brieﬂy summarise the main notation that is repeatedly used throughout this thesis.

2.1.1 Real Numbers, Vectors and Matrices

Denote the set of natural numbers by N and set N = N 0 . For the real numbers, we 0 ∪ { } use the symbol R and for a, b R , with a < b, we denote the open interval ∈ ∪ {−∞ ∞} and the closed interval by (a, b)= x R : a

d x y = xiyi and x y = (xiyj)i,j ,...,d · ⊗ =1 Xi=1 d k×l for x = (x ,...,xd),y = (y ,...,yd) R , respectively. For k, l 1, denote by R the 1 1 ∈ ≥ vector space of real (k l)-matrices. It is equipped with the scalar product × k l A : B = aijbij Xi=1 jX=1 k×l for A = (aij)i ,...,k,j ,...,l,B = (bij)i ,...,k,j ,...,l R . Denote by Ab = ((Ab)i)i ,...,k =1 =1 =1 =1 ∈ =1 k×l the matrix-vector product of a matrix A = (aij)i ,...,k,j ,...,l R and a vector b = =1 =1 ∈

19 Chapter 2 Mathematical Background

l (bj)j=1,...,l R , where ∈ l (Ab)i = aijbj. jX=1 Without distinguishing between vectors and matrices, we denote the Euclidean norm by , that is, | · | x = √x x if x Rd and A = √A : A if A Rk×l. | | · ∈ | | ∈ Moreover, we designate by det(A) the determinant of a quadratic matrix A Rk×k and for ∈ k×l t l×k B = (bij)i ,...,k,j ,...,l R , we denote by B = (bji)i ,...,k,j ,...,l R the transpose =1 =1 ∈ =1 =1 ∈ of B.

2.1.2 Topological Notions

For an arbitrary set E Rd, we designate by int(E), E and ∂E the interior, the closure ⊂ and the (topological) boundary of E, respectively. The d-dimensional open ball Br(x) d d with radius r > 0 and centre x R is deﬁned by Br(x) = y R : y x < r and ∈ { ∈ | − | } the (d 1)-dimensional unit sphere Sd−1 is given by Sd−1 = ∂B (0) = x Rd : x = 1 . − 1 { ∈ | | } Deﬁne by dist(x, E) = inf x y : y E the distance of x to E. Furthermore, we use | − | ∈ the notation F E to indicate that the closure F of a set F is compactly contained in ⊂⊂ E, i.e., F is compact and F E. ⊂

2.1.3 Diﬀerential Operators

In our notation we will not distinguish between different concepts of differentiation. The precise sense of the operators introduced below will always be clear from the context. This means that we will use the same notation for derivatives in the classical or the Sobolev sense or any other weak concept of differentiation such as distributional derivatives. For a function u:Ω R defined on an open set Ω Rd, we shall use the following → ⊂ differential operators. In dimension d = 1, u′ denotes the derivative of u and if d is arbitrary and i = 1,...,d, then ∂ u = ∂u stands for the partial derivative with respect to i ∂xi the i-th variable. Moreover, we denote by u and ∆u the gradient and the Laplacian of ∇ d u, respectively. For vector-valued functions u = (u , u ,...,ud):Ω R , the divergence 1 2 → is defined by d div(u)= ∂iui = ∂ u + ∂ u + + ∂dud 1 1 2 2 · · · Xi=1 and the vector-valued versions of the gradient and the Laplacian are then defined by

u = ∂iuj and ∆u = ∆ui = ∆u , ∆u ,..., ∆ud . ∇ i,j=1,...,d i=1,...,d 1 2 Furthermore, the symmetric (part of the) gradient Du is given by

1 t 1 Du = u + ( u) = ∂iuj + ∂jui 2 ∇ ∇ 2 i,j=1,...,d

20 2.1 Notation and by (u )u we designate the convective term which is deﬁned by ·∇ d (u )u = ui∂iuj = u1∂1uj + u2∂2uj + + ud∂duj . ·∇ j=1,...,d · · · j=1,...,d Xi=1 d×d For any matrix-valued function u = (uij)i,j ,...,d :Ω R , we deﬁne its divergence by =1 →

div(u)= ∂1u1j + + ∂dudj . · · · j=1,...,d Throughout this thesis we use the usual convention that, for time-dependent functions u = u(x,t), all the aforementioned diﬀerential operators only apply to the spatial variables.

The partial derivative of u with respect to the time variable t will be denoted by ∂tu. For simplicity, we use the same symbol for real- and vector-valued functions.

2.1.4 Normed Vector Spaces

∗ For a real normed vector space Y , we write Y for the associated norm. By Y , we k · k designate the corresponding dual space equipped with the dual norm

∗ ∗ ∗ ∗ y Y ∗ = sup y ,y Y for y Y , k k kykY ≤1 |h i | ∈

∗ where , Y : Y Y R denotes the duality pairing given by h · · i × →

∗ ∗ ∗ ∗ y ,y Y = y (y) for y Y , y Y. h i ∈ ∈

For natural numbers d, k, l N, we denote by Y d and Y k×l the vector- and the matrix- ∈ valued versions of Y which are equipped with the norms

1 1 d 2 k l 2 2 2 y Y d = yi Y and Y Y k×l = yij Y k k k k ! k k  k k  Xi=1 Xi=1 jX=1   d k×l for y = (y ,...,yd) Y and Y = (yij)i ,...k,j ,...l Y . Furthermore, we identify 1 ∈ =1 =1 ∈ Y Y 1 and Y d Y 1×d. ≃ ≃

2.1.4.1 Notions of Convergence

∗ ∗ For sequences (ym)m N Y and (y )m N Y , we use the following notions of conver- ∈ ⊂ m ∈ ⊂ ∗ ∗ gence. Let y Y and y Y . We say that ym (strongly) converges to y in Y as m if ∈ ∈ →∞ ym y Y 0 as m . We then write ym y in Y . To shorten notation, we will often k − k → →∞ → skip the supplement “strongly”. We say that ym weakly converges to y in Y as m if, →∞ ∗ ∗ ∗ for any y Y , there holds y ,ym y Y 0 as m . We then write ym⇀y weakly ∈ h − i → →∞ ∗ ∗ ∗ ∗ in Y . We say that y weakly- converges to y if y y ,y Y 0 as m holds for m ∗ h m − i → →∞ any y Y . We then write y∗ ⇁y∗ ∗ weakly- in Y ∗. Likewise, we use an analogous notation ∈ m ∗ ∗ for ordered families such as (yε)ε>0 and (yε )ε>0.

21 Chapter 2 Mathematical Background

2.1.4.2 Embeddings

We say that Y is (continuously) embedded into the normed vector space Z and denote Y ֒ Z provided that Y is a vector subspace of Z and that there exists a constant C > 0 → such that

y Z C y Y for all y Y, k k ≤ k k ∈ i.e., the (natural) inclusion operator I : Y Z given by I(y) = y is continuous. If → additionally, I is a compact operator, then the embedding is said compact, which is denoted by Y ֒ ֒ Z. By abuse of notation, we sometimes weaken the requirement that I is the →→ inclusion map y y in the sense that we replace I by certain canonical transformation of 7→ Y into Z. This is customary, for instance for embeddings of Sobolev spaces into spaces of continuous functions.

2.1.5 Function Spaces

We shall brieﬂy summarise our notation for spaces of continuous functions, Lebesgue and Sobolev spaces as well as the generalisation of the latter spaces in form of the Bochner spaces.

2.1.5.1 Spaces of Continuous Functions

For an open or a closed set S Rd, we denote the set of all continuous functions f : S R ⊂ → by C0(S). For the subspace of k-times continuously diﬀerentiable1 functions, k N, we ∈ write Ck(S). In the case k = , we use the usual deﬁnition ∞ ∞ C∞(S)= Cl(S). l\=1 Moreover, we designate

Ck([0, )) = u: [0, ) R : u = U , U Ck(R) . ∞ { ∞ → |[0,∞) ∈ } Let S Rd be either an open set or the closure of an open set. The support of f C0(S) ⊂ ∈ is given by supp(f) = x S : f(x) = 0 . With the help of the latter deﬁnition, for { ∈ 6 } k N , we introduce the space ∈ 0 ∪ {∞} Ck(S)= f Ck(S) : supp(f) S 0 { ∈ ⊂⊂ } k ∞ of compactly supported C functions. By C0(S), we denote the closure of C0 (S) with respect to the norm

d u ∞ = sup u(x) for u C0(S) or for u C0(S) . k k x∈S | | ∈ ∈

1Here and subsequently, requiring that a (partial) derivative is continuous in a non-open set Q ⊂ Rd means that it exists in int(Q) and has a continuous extension to Q. By abuse of notation, we do not distinguish between the derivative and its extension.

22 2.1 Notation

2.1.5.2 Functions Depending on Time

For an arbitrary set Q Rd, we denote by C0(Q) the space of all continuous functions ⊂ u: Q R and define C0(Q) as the subspace of all bounded functions u C0(Q). We → b ∈ define C∞ (Q)= u: Q R : u = U , U C∞(Rd), supp(u) Q . (0) { → |Q ∈ 0 ⊂ } ∞ Note that if Q is not an open set, then functions belonging C(0)(Q) do not necessarily vanish near the boundary ∂Q. For example, if Q = Ω [0, T ) for an open set Ω Rd−1, × ⊂ d 2, and some T > 0, then, for ϕ C∞ (Ω [0, T )), it does not hold ϕ(x, 0) = 0 for ≥ ∈ (0) × x Ω, in general. ∈ To shorten notation, we introduce the multi-index notation. For π Nd, we denote ∈ 0 π π π π d = ∂ 1 ∂ 2 ...∂ d and π = π + π + + πd. For an open set Ω R and T > 0, ∇ x1 x2 xd | |∗ 1 2 · · · ⊂ define C1(Ω [0, T )) by ×

0 π 0 d u C (Ω [0, T )) : ∂tu, u C (Ω [0, T )), π N , π = 1 . { ∈ × ∇ ∈ × ∈ 0 | |∗ }

For functions with diﬀerent regularity properties with respect to the time variable and the spatial variables, for k N, we introduce the space Ck,1(Ω (0, T )) deﬁned by ∈ ×

u C0(Ω (0, T )) : ∂s πu C0(Ω (0, T )), 1 2s + π k, s N , π Nn . { ∈ × t ∇ ∈ × ≤ | |∗ ≤ ∈ 0 ∈ 0 }

For a general subset Q of Rd [0, T ], in a similar fashion, we deﬁne the space C3,1(Q) by × b

u C0(Q) : ∂s πu C0(Q), 1 2s + π 3, s N , π Nn . { ∈ b t ∇ ∈ b ≤ | |∗ ≤ ∈ 0 ∈ 0 }

2.1.5.3 Distributions

For an open set S Rd, set (S) C∞(S). (S) is a linear space equipped with the ⊂ D ≡ 0 D following notion of continuity. Let ϕm, m N, and ϕ belong to (S). We say that ϕm ∈ D converges to ϕ in (S) as m if there exists some compact set K S such that D → ∞ ⊂⊂ d π π supp(ϕm), supp(ϕ) K for all m N and, for any π N , there holds ϕm ϕ ⊂ ∈ ∈ 0 ∇ → ∇ uniformly on K if m . →∞ A distribution is a continuous linear mapping

f : (S) R, ϕ f, ϕ = f, ϕ , D → 7→ h iD(S) h i i.e., if ϕm ϕ in (S) as m , then there holds → D →∞

lim f, ϕm S = f, ϕ S . m→∞h iD( ) h iD( )

We designate the set of all distributions by ′(S). For further information about distribu- D tions we refer to [83, Chapter 5].

23 Chapter 2 Mathematical Background

2.1.5.4 Lebesgue and Sobolev Spaces

Let M Rd be a (Lebesgue-)measurable set and let r [1, ]. Throughout this thesis ⊂ ∈ ∞ we denote the d-dimensional Lebesgue measure by d, write M = d(M) and denote by L | | L dx integration with respect to d. For a (Lebesgue-)measurable function u: M R, we L → deﬁne the (possibly inﬁnite) Lebesgue norm u r by k kL (M)

1 r r ( M u(x) dx) if r< , u Lr (M) =  | | ∞ k k essR sup u(x) if r = ,  x∈M | | ∞  where ess sup stands for the essential supremum. Then the Lebesgue spaces Lr(M) consists of all measurable functions u: M R such that u r is ﬁnite. → k kL (M) r Additionally, we deﬁne the space Lloc(M) of locally integrable functions by

r r L (M)= u: M R : uχK L (M) for all K M . loc { → ∈ ⊂⊂ }

For a domain D Rd, we deﬁne the Sobolev space W k,r(D) of order k N and integrabil- ⊂ ∈ ity exponent r by

W k,r(D)= u Lr(D) : πu Lr(D) for all π Nd with π k . { ∈ ∇ ∈ ∈ | |∗ ≤ }

Here, the symbol π stands for diﬀerentiation in the weak (or Sobolev) sense. The space ∇ W k,r(D) is equipped with the norm

1 2 π 2 u k,r = u r . k kW (D)  k∇ kL (D) |πX|∗≤k   k,r ∞ k,r By W0 (D), we denote the closure of C0 (D) in W (D). For notational convenience, we 0,s s k k,2 k k,2 set W (D)= L (D), H (D)= W (D) and H0 (D)= W0 (D). ′ −1,r 1,r The Sobolev space W (D) of negative order is deﬁned as the dual space of W0 (D), ′ −1,r 1,r ∗ 1 1 i.e., we have W (D)= W0 (D) with r + r′ = 1 and the customary convention that 1 −1 −1.2 ∞ = 0. In the case r = 2, we abbreviate H (D)= W (D). We denote the trace of a Sobolev function u on a manifold ω by u whenever it is well- |ω deﬁned. For convenience, often we will simply write u instead of u , where no confusion |ω arises. The corresponding Lebesgue spaces will be denoted by Lr(ω). In the case that ω is the boundary of a domain D of class C1, i.e., M = ∂D, we designate by

1− 1 ,r 1,r W r (∂D)= u : u W (D) |∂D ∈ 1− 1 ,r the range of the trace operator. We equip the space W r (∂D) with the norm

1 r r u(x) u(y) d−1 d−1 1 r u 1− ,r = u L (∂D) + | −r+d−1| d (x)d (y) , k kW r (∂D) k k ∂D ∂D x y H H ! Z Z | − |

24 2.1 Notation where d−1 denotes the (d 1)-dimensional Hausdorff measure, see Definition 2.2.14 below, H − 1 1 ,2 and use the notation H 2 (∂D)= W 2 (∂D) for r = 2. Not merely for the trace operator, we will make use of the standard theory for Sobolev spaces: for example, common facts about extension operators, embedding properties into Lebesgue spaces and spaces of continuous functions as well as density results will be used. Furthermore, we will apply standard inequalities such as the inequalities of Young, Hölder, Poincaréand related estimates such as interpolation inequalities. For precise formulations and proofs of all these results we refer to standard text books such as [8, 19].

2.1.5.5 Bochner Spaces

The Bochner spaces extend the concept of the Lebesgue spaces Lr(I), where I R is ⊂ some interval, to functions u: I Y taking values in some Banach space Y . Throughout → this thesis the interval I will always be a subset of the time interval [0, ). The deﬁnition ∞ of the Bochner spaces is based on the following notions of measurability and integrability.

Deﬁnition 2.1.1 (Bochner measurability). Let I R be an interval and let Y be a ⊂ Banach space.

1. A function s: I Y is called a simple function if, for some k N, there exist pairwise → ∈ disjoint sets B ,B ,...,Bk I with ﬁnite Lebesgue measure and y ,y ,...,yk Y 1 2 ⊂ 1 2 ∈ such that k

s(t)= yiχBi (t) Xi=1 for all t I, where χBi : I R denotes the characteristic function of Bi, that is, ∈ →

1 if t Bi, ∈ χBi (t)=  0 if t / Bi.  ∈  2. A function u: I Y is said to be Bochner measurable if there exists a sequence (sm)m N → ∈ of simple functions sm : I Y such that, for a.e. t I, there holds → ∈

lim sm(t) u(t) Y = 0. m→∞ k − k

Deﬁnition 2.1.2 (Bochner integrability). Let I R be an interval and let Y be a Banach ⊂ space. A function u: I Y is called Bochner integrable if there exists a sequence (sm)m N → ∈ of simple functions sm : I Y with →

lim sm(t) u(t) Y dt = 0. m→∞ k − k ZI Then, for any Bochner measurable function u: I Y , the corresponding Bochner integral → is given by

u(t)dt = lim sm(t)dt, m→∞ ZI ZI where the limit is taken in Y .

25 Chapter 2 Mathematical Background

With the help of the latter definition we may define the Bochner space L1(I; Y ) as the space of all Y -valued Bochner integrable functions u: I Y . For r [1, ], we define → ∈ ∞ the (possibly infinite) norm

1 r r ( I u(t) Y dt) , if r< , u Lr(I;Y ) =  k k ∞ k k R ess sup u(t) Y , if r = .  t∈I k k ∞  The corresponding Bochner spaces are given by

r 1 L (I; Y )= u L (I; Y ) : u r < . { ∈ k kL (I;Y ) ∞}

For notational convenience, for open intervals (a, b), we set Lr(a, b; Y )= Lr((a, b); Y ). For more detailed information on Bochner spaces we refer for example to [99].

2.2 Measures

The purpose of this section is to brieﬂy recall some deﬁnitions and facts about measures. For more detailed information we refer to the text books [16, 19, 39].

2.2.1 Radon Measures

In what follows, let X be a locally compact, separable metric space such as Rd, d N, or ∈ its open or closed subsets. Let us recall some measure-theoretic notions for set functions µ: (X) Rs, where (X) is the Borel σ-algebra of X, i.e., the σ-algebra generated by B → B the open subsets of X, and s N is an arbitrary dimension. ∈

Definition 2.2.1 (Radon measure). A set function µ: (X) Rs is called Rs-valued B → finite Radon measure (or, briefly, Radon measure) if µ is a measure on (X); that is, B µ( ) = 0 and µ is σ-additive, i.e., for all pairwise disjoint families (Em)m N (X), ∅ ∈ ⊂ B there holds ∞ ∞ µ Em = µ(Em). m ! m [=1 X=1 The space of all Rs-valued finite Radon measures is designated by (X)s. Additionally, M define (X)= (X)1 and M M

+(X)= µ (X) : µ(B) 0 for all B (X) . M { ∈ M ≥ ∈B }

Deﬁnition 2.2.2 (Measurability and integration). 1. A function f : X Rs is called → (X)-measurable (or, brieﬂy, measurable) if, for any open set O Rs, there holds B ⊂

f −1(O)= x X : f(x) O (X). { ∈ ∈ }∈B

26 2.2 Measures

2. Let µ +(X). For any B (X), denote by ∈ M ∈B

f dµ = χBf dµ ZB ZX the integral of a measurable function f : X Rs with respect to µ and designate → L1(X,µ)= f : X R : f measurable, f dµ< . → X | | ∞ n Z o 3. Let B (X). For f L1(X,µ)s and µ +(X), set ∈B ∈ ∈ M

f dµ = f1 dµ,..., fs dµ B B B Z Z Z Likewise, deﬁne

f dµ = f dµ1,..., f dµs B B B Z Z Z for f L1(X,µ) and µ +(X)s and ∈ ∈ M

f dµ = f dµ + + fs dµs 1 1 · · · ZB ZB ZB for f L1(X,µ)s and µ +(X)s. ∈ ∈ M Definition 2.2.3. Let µ +(X), E (X) and f L1(X,µ)s. ∈ M ∈B ∈ 1. The restriction measure µxE : (X) R is defined as B → µxE(B)= µ(E B) for B (X). ∩ ∈B 2. Define fµ: (X) Rs by B → fµ(B)= f dµ for B (X). ∈B ZB The total variation of a Radon measure is defined as follows.

Deﬁnition 2.2.4 (Total variation). Let µ (X)s. Then the total variation of µ in a ∈ M set E (X) is deﬁned by ∈B ∞ µ (E) = sup µ(Em) , | | (m | |) X=1 where the supremum is taken over all pairwise disjoint partitions (Em)m N X of meas- ∈ ⊂ ∞ urable sets Em, m N, such that E = Em. ∈ m=1 Theorem 2.2.5. For every Radon measureS µ (X), the total variation µ is a non- ∈ M | | negative Radon measure, that is, µ +(X). | | ∈ M Proof. See [16, Theorem 1.6].

The notion of total variation allows one to equip the space (X)s with a norm. M s Theorem 2.2.6 (Total-variation norm). The mapping s : (X) R given by k · kM(X) M →

µ µ s = µ (X), (2.2.1) 7→ k kM(X) | | s s s for µ (X) , deﬁnes a norm on (X) . Moreover, the space ( (X) , s ) is ∈ M M M k · kM(X) a Banach space.

Proof. See [97, p. 21–23].

27 Chapter 2 Mathematical Background

2.2.1.1 Representation as Dual Space

Here and subsequently, let C0(X) denote the closure of the real-valued continuous functions on X with compact support on X with respect to the norm

η ∞ = sup η(x) . k k x∈X | | Every Radon measure µ (X)s allows one to deﬁne the pairing ∈ M s µ, η s = η dµ for η C (X) . (2.2.2) h iC0(X) ∈ 0 ZX The mapping η µ, η s is linear and satisﬁes the estimate 7→ h iC0(X)

µ, η s = η dµ µ(X) η . h iC0(X) ≤ | | k k∞ ZX

Thus, by the deﬁnition of total variation, we have

µ, η s µ (X) η = µ s η . h iC0(X) ≤ | | k k∞ k kM(X) k k∞

s This means that the linear mapping µ, s : (C (X) , ) R is continuous. h ·iC0(X) 0 k · k∞ → s ∗ Hence, via (2.2.2), µ generates an element of the dual space (C0(X) ) . s ∗ Conversely, every element of (C0(X) ) can be represented by a Radon measure in the above manner. Indeed, the following representation theorem allows one to isometrically identify the spaces (X)s and (C (X)s)∗. M 0 Theorem 2.2.7 (Riesz representation theorem). For every L (C (X)s)∗, there exists a ∈ 0 s s unique µL (X) such that µL, η s = L(η) for all η C (X) , and there holds ∈ M h iC0(X) ∈ 0 s µL s = sup L(η) : η C (X) , η 1 . k kM(X) | | ∈ 0 k k∞ ≤ n o Proof. See [16, Proposition 1.47 and Theorem 1.54].

2.2.1.2 Weak- Convergence and Compactness ∗ We recall the notion of weak- convergence on (Ω)s and provide its compactness prop- ∗ M erties.

s s Deﬁnition 2.2.8 (Weak- convergence). A sequence (µm)m N (X) of ﬁnite R - ∗ ∈ ⊂ M valued Radon measures is said to converge weakly- to µ (X)s as m , written ∗ ∈ M → ∞ ∗ s s µm ⇁µ in (X) , if, for any ϕ C (X) , there holds k M ∈ 0

lim ϕ dµm = ϕ dµ. m→∞ ZX ZX s Theorem 2.2.9 (Weak- compactness). Let (µm)m N (X) be such that ∗ ∈ ⊂ M

sup µm (X) < . m∈N | | ∞ s Then there exist a subsequence (µm )k N and a limiting Radon measure µ (X) such k ∈ ∈ M ∗ s that µm ⇁µ in (X) as k . k M →∞ Proof. See [16, Theorem 1.59].

28 2.2 Measures

2.2.2 Derivation of Radon measures

We provide a result concerning the derivation of Radon measures due to Radon and Nikodym. To formulate the theorem, we need the following measure-theoretic notions.

Deﬁnition 2.2.10 (Absolute continuity and singularity). 1. Let ν +(X) and µ ∈ M ∈ +(X)s. Then ν is called absolutely continuous with respect to µ, written ν µ, if, M ≪ for any B (X) with µ (B) = 0, there holds ν(B) = 0. ∈B | | 2. µ,ν +(X) are called singular, written µ ν, if there exists a set B (X) such ∈ M ⊥ ∈ B that µ(B) = 0 and ν(X B) = 0. Likewise, µ,ν (X)s are called singular, written \ ∈ M µ ν, if µ ν . ⊥ | | ⊥ | | The Radon–Nikodym theorem generalises the fact that

µ ν µ = fν for some f L1(X,ν)s. ≪ ⇐⇒ ∈ Theorem 2.2.11 (Radon–Nikodym). Let µ (X)s and ν +(X). Then there ∈ M ∈ M exists a unique pair (µ ,µ ) (X)s (X)s such that µ ν, µ ν and ac sing ∈ M × M ac ≪ sing⊥ µ = µ +µ . Moreover, there exists a unique function f L1(X,ν)s such that µ = fν. ac sing ∈ ac The function f is called the Radon–Nikodym derivative of µ with respect to ν and it is denoted by µ . For a.e. x X, it is given by ν ∈ µ µ(B (x)) (x)= f(x) = lim δ . ν δց0 ν(Bδ(x)) Proof. See [16, Theorem 1.28] or [19, Theorem 4.2.1].

µ Throughout this thesis we only need the special case of the Radon–Nikodym derivative |µ| of µ with respect to its total variation µ . Note that, by the deﬁnition of total variation, | | it always holds µ µ . ≪ | | Corollary 2.2.12 (Polar decomposition). For any µ (X)s, there exists a unique ∈ M function f L1(X,ν)s with f(x) = 1 for µ-a.e. x X such that µ = f µ . ∈ | | ∈ | | Proof. See [16, Corollary 1.29].

Functionals of the form µ µ x, |µ| (x) d µ (x) 7→ X F | | Z have the following continuity property with respect to weak- convergence. ∗ d s Theorem 2.2.13 (Reshetnyak continuity). For an open set X R , let µm,µ (X) , ⊂ ∈ M ∗ s m N, be such that µm ⇁ µ in (X) and µm (X) µ (X) for m . Then, for ∈ M | | → | | → ∞ every continuous and bounded function : X Sd−1 R, there holds F × →

µm µ lim x, |µ | (x) d µm (x)= x, |µ| (x) d µ (x). m→∞ X F m | | X F | | Z Z Proof. See [16, Theorem 2.39].

29 Chapter 2 Mathematical Background

2.2.3 Hausdorﬀ Measure

We brieﬂy introduce the (d 1)-dimensional Hausdorﬀ measure d−1, d 2. For its − H ≥ further properties we refer for example to [19, Section 4.1] or [39, Chapter 2].

Definition 2.2.14 (Hausdorff measure). Let diam(B) = x y : x,y Rd designate {| − | ∈ } d d−1 the diameter of a set B R and let ωd = (B (0)) denote the volume of the ⊂ −1 L 1 (d 1)-dimensional unit ball. For δ > 0 and E Rd, set − ⊂ ∞ ∞ d−1 ωd−1 d−1 δ (E)= d inf diam(Am) : E Am, diam(Am) < δ . H 2 −1 ( ⊂ ) mX=1 m[=1 The (d 1)-dimensional Hausdorff measure d−1(E) of E is defined by − H

d−1 d−1 (E) = lim δ (E). H δց0 H

Remark 2.2.15 (Surface measure). Whenever E is a Borel-measurable subset of Rd that is contained in a (d 1)-dimensional C1-manifold, the classical surface measure of E is − equal to d−1(E); see [16, Theorem 2.71]. We want to point out that this means that we H may express the classical Gauß–Green theorem by

div(u)dx = u ν d d−1(x), · H ZΩ Z∂Ω where Ω Rd is a bounded domain with C1-boundary ∂Ω and outer unit normal ν, and ⊂ u C1(Ω)d. Later, in Theorem 2.3.13, we will generalise the latter formula to less regular ∈ sets Ω. This generalised Gauß–Green formula will turn out to be appropriate for our purposes.

2.3 Functions of Bounded Variation

Treating two-phase flow problems one is naturally faced with functions having discontinuities (jumps) along one-codimensional manifolds. Their first order distributional derivatives are measures which may charge sets of Lebesgue measure zero. These functions therefore do not belong to classical Sobolev spaces, such as W 1,1, in general. The right framework for our purpose is the “slightly larger” space of functions of bounded variation. Here, we briefly introduce this space. For more detailed information we refer to the text books [16, 19, 39].

2.3.1 Deﬁnitions and Basic Properties

In the remainder of this section let Ω Rd, d N, be an open set. ⊂ ∈ Deﬁnition 2.3.1 (Functions of bounded variation). A function u L1(Ω) is said to be ∈ of bounded variation if its distributional gradient u is a Radon measure; that is, there ∇

30 2.3 Functions of Bounded Variation exists some µ (Ω)d such that, for all ϕ C∞(Ω) and any i = 1,...,d, there holds ∈ M ∈ 0

ϕ dµi = u ∂iϕ dx. − · ZΩ ZΩ The set of all functions of bounded variation is denoted by BV (Ω) and the set BV (Ω, M) contains all functions u BV (Ω) such that u M for a.e. x Ω. ∈ ∈ ∈ Proposition 2.3.2 (Total variation). For any u BV (Ω), there holds ∈ 1 d u d = u (Ω) = sup u div(ψ)dx : ψ C (Ω) , ψ 1 . k∇ kM(Ω) |∇ | ∈ 0 k k∞ ≤ ZΩ Proof. See [39, p. 170].

To deﬁne an appropriate norm on the space BV (Ω), we use the total-variation norm

d introduced in (2.2.1). k · kM(Ω) Theorem 2.3.3. The space BV (Ω) is a Banach space equipped with the norm

u = u 1 + u d . (2.3.1) k kBV (Ω) k kL (Ω) k∇ kM(Ω) Proof. See [19, Theorem 10.1.1].

Remark 2.3.4 (Generalisation of Sobolev functions). For any u W 1,1(Ω), the distribu- ∈ tional derivative is given by u d, where d denotes the d-dimensional Lebesgue measure, ∇ L L and thus the inclusion W 1,1(Ω) BV (Ω) holds true; see for example [16, p. 118] or [39, ⊂ 5.1, Example 1]. Hence the BV -norm (2.3.1) is an extension of the W 1,1(Ω)-norm to 1,1 the space BV (Ω). Indeed, for all u W (Ω), there holds u = u 1,1 . In ∈ k kBV (Ω) k kW (Ω) .particular, the embedding W 1,1(Ω) ֒ BV (Ω) is continuous →

2.3.2 Compactness Properties

We provide the following two results concerning the embedding properties of BV (Ω) into Lebesgue spaces and the lower semi-continuity with respect to strong L1-convergence.

∗ ∗ d Theorem 2.3.5 (Embedding theorem). Let 1 = if d = 1 and 1 = d−1 if d 2. If ∞ ∗ ≥ Ω is bounded with Lipschitz boundary ∂Ω, the embedding BV (Ω) ֒ L1 (Ω) is continuous → .(∗and the embedding BV (Ω) ֒ ֒ Lp(Ω) is compact for all p [1, 1 →→ ∈ Proof. See [16, Proposition 3.21 and Corollary 3.49].

1 Theorem 2.3.6 (Lower semi-continuity). If (um)m N BV (Ω) converges in L (Ω) for ∈ ⊂ m to some limit function u L1(Ω), then there holds →∞ ∈

u (Ω) lim inf um (Ω). |∇ | ≤ m→∞ |∇ | In particular, u BV (Ω) if additionally, there holds ∈

sup um BV (Ω) < . m∈N k k ∞ Proof. See [19, Proposition 10.1.1] or [39, Section 5.2.1, Theorem].

31 Chapter 2 Mathematical Background

2.3.3 Sets of Finite Perimeter

We have already seen in Remark 2.3.4 that Sobolev function are of bounded variation. Now

we shall consider functions of the special form u = χE, where χE denotes the characteristic (or indicator) function of a set E Ω, i.e., ⊂ 1 for x E, χE(x)=  ∈ 0 for x Ω E.  ∈ \ Of particular interest are the corresponding distributional derivatives u = χE. Note  ∇ ∇ 1,1 that, in general, χE does not belong to the Sobolev space W (Ω). To relate properties

of E to properties of χE we give the following deﬁnition.

Definition 2.3.7 (Sets of finite perimeter). A measurable set E Rd has finite perimeter ⊂ in Ω if its characteristic function χE belongs to BV (Ω). The perimeter (E, Ω) is defined P by (E, Ω) = χE (Ω), i.e., by the total variation of the distributional gradient χE in P |∇ | ∇ Ω.

Remark 2.3.8. Let E Ω be a set of ﬁnite perimeter in Ω. Then applying Proposi- ⊂ tion 2.3.2 to u = χE yields

1 d (E, Ω) = χE (Ω) = sup div(ψ)dx : ψ C (Ω) , ψ 1 . P |∇ | ∈ 0 k k∞ ≤ ZE The following example shows that the class of sets of ﬁnite perimeter includes all subsets with Lipschitz boundaries.

Example 2.3.9 (Lipschitz sets). Let E be an open, bounded subset of Ω with Lipschitz boundary ∂E. Then E has ﬁnite perimeter in Ω and there holds (E, Ω) = d−1(∂E Ω); P H ∩ see [19, Remark 10.3.3]. Thus the perimeter of E in Ω measures the “size” of ∂E in Ω.

The foregoing example is important as it indicates that sets of finite perimeter are more general than sets with Lipschitz boundaries. For open, bounded Lipschitz domains E Ω, ⊂ d−1 the outer unit normal νE = νE(x) is meaningful for -a.e. x ∂E and the Gauß–Green H ∈ formula d−1 div(u)dx = u νE d (x) (2.3.2) · H ZE Z∂E holds true for Sobolev fields u W 1,1(E)d, cf. [39, 4.3, Theorem 1]. As a direct con- ∈ sequence, cf. Remark 2.2.15, it follows that E has finite perimeter in Ω. In particular, in d−1 (2.3.2), the outer normal νE = νE(x) is meaningful for -a.e. x ∂E, and there holds H ∈ d−1 d−1 χE (Ω) = x∂E and χE = νE x∂E = νE χE (Ω); (2.3.3) |∇ | H ∇ H |∇ | see [39, p. 171]. We shall generalise the Gauß–Green formula as well as (2.3.3) to sets E of finite perimeter,

i.e., to characteristic functions χE. To this end, we introduce two measure-theoretic concepts of boundary, namely, the reduced and the essential boundary, that allow us to extend the notions of outer and inner normal to sets E that are merely of ﬁnite perimeter.

32 2.3 Functions of Bounded Variation

2.3.4 The Generalised Gauß–Green Theorem

In the following deﬁnition, we introduce a measure-theoretic notion of unit normal to a set of ﬁnite perimeter which naturally leads to the notion of reduced boundary.

Deﬁnition 2.3.10 (Generalised normal and reduced boundary). Let E Rd be a set of ⊂ ﬁnite perimeter in Ω.

1. The generalised unit inner normal ν to E is the Radon–Nikodym derivative of χE i ∇ with respect to the measure χE . This means, for χE -a.e. x Ω, |∇ | |∇ | ∈ χE χE(Bδ(x)) νi(x)= ∇ = lim ∇ . χE δց0 χE (Bδ(x)) |∇ | |∇ | Likewise, ν = ν is the generalised unit outer normal ν to E. o − i o 2. The reduced boundary ∂∗E is the set of all x Ω such that the above limit exists and ∈ ν (x) = 1. | i | Using the topological notions of interior and boundary, a set E provides a disjoint partition of Ω via Ω = (int(E) Ω) (∂E Ω) int(Ω E). The notion of essential boundary ∂ E ∩ ∪ ∩ ∪ \ ∗ provides a measure-theoretic analogue Ω = (E0 Ω) (∂ E Ω) (E1 Ω) in the sense ∩ ∪ ∗ ∩ ∪ ∩ of the following deﬁnition.

Deﬁnition 2.3.11 (Essential boundary). For a measurable set E Rd and s [0, 1], ⊂ ∈ deﬁne the set of all points of density s by

E Bδ(x) Es = x Rd : lim | ∩ | = s . ∈ δց0 Bδ(x) | | 1. The sets E0 and E1 are called the measure-theoretic exterior and interior of E, respectively.

2. The essential boundary ∂ E is deﬁned as ∂ E = Rd (E E ). ∗ ∗ \ 0 ∪ 1 The following theorem due to Federer allows one to compare the notions of the reduced and the essential boundary.

Theorem 2.3.12 (Federer). For a set E Rd of ﬁnite perimeter in Ω, the following ⊂ statements are valid.

∗ 1 1. ∂ E Ω E 2 ∂ E. ∩ ⊂ ⊂ ∗ 2. d−1 Ω (E0 ∂∗E E1) = 0. H \ ∪ ∪ 0 1 1 d−1 In particular, there holds x E E 2 E for -a.e. x E, and ∈ ∪ ∪ H ∈ d−1 Ω (∂ E ∂∗E) = 0. H ∩ ∗ \ Proof. See [16, Theorem 3.61].

33 Chapter 2 Mathematical Background

Theorem 2.3.13 (Generalised Gauß–Green theorem). Let E Rd be set of ﬁnite peri- ⊂ meter in Ω. Then the corresponding characteristic function χE belongs to BV (Ω) and, for

its distributional gradient χE, there holds χE = νE χE , where νE is the generalised ∇ ∇ |∇ | unit inner normal to E, and

d−1 ∗ (E, Ω) = χE (Ω) = (∂ E Ω). (2.3.4) P |∇ | H ∩ Moreover, for all ψ C1(Ω)d, there holds ∈ 0

d−1 div(ψ)dx = ψ νE d (x). (2.3.5) − ∗ · H ZE Z∂ E Proof. See [19, Theorem 10.3.2], [16, Theorem 3.36] or [39, 5.8, Theorem 1].

Remark 2.3.14. Theorem 2.3.12 allows us to replace the reduced boundary ∂∗E in (2.3.4) 1 2 1 and (2.3.5) by either the essential boundary ∂∗E or the set E of points of density 2 .

34 Chapter 3

The Diﬀuse-Interface Model

In this chapter we are concerned with an appropriate weak formulation for the phase-ﬁeld- like scaling of the Navier–Stokes–Korteweg model (1.4.8). In the preliminary Section 3.1 we will study the double-well potential W ; see (1.4.5). In Section 3.2 we will derive the weak formulation. For families of weak solutions (depending on the parameter ε), we shall prove a priori estimates in Section 3.3, and in Section 3.4 we will establish appropriate compactness properties. We will justify the claim that these compactness properties yield a sharp interface dividing the domain into two phases in Section 3.5. Later, in Chapter 5, the results of Sections 3.4 and 3.5 will be used to perform the sharp-interface limit for (1.4.8). Therefore, we will review the compactness properties of weak solutions in regard to the sharp-interface limit in the concluding Section 3.6.

3.1 The Double-Well Potential

We shall state regularity assumptions on the normalized double-well potential W introduced in (1.4.5). Moreover, we provide a growth condition that, on the one hand, ensures the double-well shape of W , and, on the other hand, allows for analytical treatment.

Assumptions 3.1.1 (Double-well potential). Suppose that the double-well potential W as in (1.4.5) has the following properties.

1. Regularity. W : [0, ) R belongs to C2([0, )). ∞ → ∞ 2. Non-negative function. W is non-negative, i.e., W (z) 0 for all z [0, ). ≥ ∈ ∞ 3. Exactly two zeros/global minimal points. W has exactly two zeros at some points β , β [0, ), such that 0 < β < β , i.e., W (z) = 0 if and only if z β , β . 1 2 ∈ ∞ 1 2 ∈ { 1 2}

β1+β2 β2−β1 4. Growth condition. Let a = 2 and b = 2 . There exist constants C1 > 0 and C (0, b) such that, for some p > 2, there holds 2 ∈ ∗ W ′′(z) C z a p∗−2 for all z [0, ) with z a b C . (3.1.1) ≥ 1 | − | ∈ ∞ | − |≥ − 2

Remark 3.1.2 (Growth of the double-well potential). The growth condition (3.1.1) is a classical assumption, cf. for example [29, equation (1.2)], [76, equation (2.15)] or [92,

35 Chapter 3 The Diﬀuse-Interface Model

1 equation (1.32)]. It implies strong L (Ω)-compactness of the sequence (ρε(t))ε>0 by uniform boundedness of an appropriate energy functional. We will carry out the details of this argument in Section 3.3.2.

The following canonical polynomial double-well potential satisﬁes Assumptions 3.1.1.

Example 3.1.3 (Canonical double-well potential). The polynomial double-well potential

W (z) = (z β )2(z β )2 = (z a)2 b2 2, (3.1.2) − 1 − 2 − − β1+β2 β2−β1 where a = 2 and b = 2 satisﬁes Assumptions 3.1.1. Indeed, we have

W ′′(z) = 12(z a)2 4b2. − −

We may choose C = 1 b and assume that z [0, ) satisﬁes z a b C = 2 b, or 2 3 ∈ ∞ | − | ≥ − 2 3 equivalently, b2 9 z a 2. Then we estimate ≤ 4 | − |

W ′′(z) = 12 z a 2 4b2 12 z a 2 9 z a 2 = 3 z a 2 . | − | − ≥ | − | − | − | | − |

Hence W satisﬁes the growth condition (3.1.1) for p∗ = 4 with constants C1 = 3 and 1 C2 = 3 b.

We collect some consequences of Assumptions 3.1.1 that will be useful for our analysis. We show that the growth condition (3.1.1) for W ′′ implies also control for W .

Lemma 3.1.4 (Growth of W ). Let the double-well potential W be as in Assumptions 3.1.1. Then there exist constants C ,C > 0 such that, for all z [0, ), the following estimates 1 2 ∈ ∞ hold true.

1. W (z) C z a p∗ C . ≥ 1 | − | − 2 2. ( z a b)2 C W (z). | − |− ≤ 1 Proof. Let z [0, ). For C as in (3.1.1), we distinguish the following three cases. ∈ ∞ 2 (i) z y [0, ) : y a b C = y [0, ) : y β C . ∈ { ∈ ∞ − ≥ − 2} { ∈ ∞ ≥ 2 − 2} (ii) z y [0, ) : y a b C = y [0, ) : β + C y β C . ∈ { ∈ ∞ | − |≤ − 2} { ∈ ∞ 1 2 ≤ ≤ 2 − 2} (iii) z y [0, ) : y a C b = y [0, ) : y β + C . ∈ { ∈ ∞ − ≤ 2 − } { ∈ ∞ ≤ 1 2} In case (ii) both estimates follow from the facts that W is strictly positive on the compact interval [β + C , β C ], and that all involved functions are continuous. Furthermore, 1 2 2 − 2 we will only give a proof in case (i), as case (iii) follows analogously. This means, without loss of generality, we will assume that z β C =: β∗. Note that then β∗ > 0 and ≥ 2 − 2 2 2 z a β∗ a = b C > 0. − ≥ 2 − − 2

36 3.2 The Notion of Weak Solutions

1. Taking into account the growth condition (3.1.1), we have z z ′ ′ ∗ ′′ p∗−2 W (z) W (β2 )= W (y)dy C1 (y a) dy. (3.1.3) − β∗ ≥ β∗ − Z 2 Z 2 Evaluating the integral on the right-hand side leads to

W ′(z) C1 (z a)p∗−1 (β∗ a)p∗−1 + W ′(β∗) ≥ p∗−1 − − 2 − 2 = D (z a)p∗−1 D 1 − − 2 for suitable constants D1, D2 > 0. Similarly to (3.1.3), we infer that z z ∗ ′ p∗−1 W (z) W (z) W (β2 )= W (y)dy D1(y a) D2 dy ≥ − β∗ ≥ β∗ − − Z 2 Z 2 since W (β∗) 0 due to Assumptions 3.1.1. Therefore, we conclude that 2 ≥ z−a p∗−1 ∗ W (z) D1 y dy D2(z β2 ) ≥ β∗−a − − Z 2 = D1 (z a)p∗ (β∗ a)p∗ D (z a) D (a β∗). p∗ − − 2 − − 2 − − 2 − 2 Remarking that D (z a) 1 D (z a)p∗ + D , for some constant D > 0, due to 2 − ≤ 2p∗ 1 − 3 3 Young’s inequality, we obtain the ﬁrst claim.

′ 2. By Assumptions 3.1.1, we infer that W (β2) = W (β2) = 0. Then, due to the Taylor series of W at β , there exists some ξ β∗ such that 2 ≥ 2 W (z)= W (β )+ W ′(β )(z β )+ 1 W ′′(ξ)(z β )2 = 1 W ′′(ξ)(z β )2. 2 2 − 2 2 − 2 2 − 2 Using the growth condition (3.1.1) and z β = z a b yields − 2 | − |− W (z) 1 C ξ a p∗−2 (z β )2 1 C b C p∗−2 ( z a b)2 . ≥ 2 1 | − | − 2 ≥ 2 1 | − 2| | − |− This completes the proof.

3.2 The Notion of Weak Solutions

In this section we are concerned with a weak formulation for the phase-ﬁeld-like scaling of the Navier–Stokes–Korteweg equations (1.4.8) with respect to prescribed boundary and initial data. More precisely, we will consider the partial diﬀerential equations

∂tρε + div(ρεvε) = 0, (3.2.1) 1 ∂t(ρεvε) + div(ρεvε vε)+ p(ρε) = 2div(µ(ρε)Dvε)+ ερε ∆ρε, (3.2.2) ⊗ ε ∇ ∇ depending on a parameter ε > 0, in the space-time cylinder Ω (0, T ). We close the × system by adding the boundary and initial conditions

ρε ν =0 on ∂Ω [0, T ), (3.2.3) ∇ · × vε =0 on ∂Ω [0, T ), (3.2.4) × (i) ρε( , 0) = ρ in Ω, (3.2.5) · ε (i) vε( , 0) = v in Ω. (3.2.6) · ε

37 Chapter 3 The Diﬀuse-Interface Model

In (3.2.2) the non-monotone pressure function p: [0, ) R is deﬁned as in Section 1.4.1, ∞ → and is related to the (normalized) double-well potential W by p′(ρ)= ρW ′′(ρ); see (1.4.5) and (1.4.6). For our analysis, we will use this identity and we shall formulate assumptions on the pressure function in terms of W .

Assumptions 3.2.1. Throughout this thesis let the following general assumptions hold true.

1. Smoothness of domain. Let Ω Rn, n = 2, 3, be a bounded domain with boundary ⊂ ∂Ω of class C2 and outer unit normal ν.

2. Regularity and boundedness of viscosity. For some constants 0 < cµ Cµ, let ≤ the viscosity function µ: [0, ) [cµ,Cµ] be bounded and Lipschitz continuous. ∞ →

3. Double-well potential. Let W be as in Assumptions 3.1.1.

In this section we will study suﬃciently smooth solutions (ρε, vε) of the diﬀuse-interface model (3.2.1)–(3.2.6) in order to derive an appropriate weak formulation. Later, for this notion of solutions, we aim to study the singular limit as ε 0 in Chapter 5. → Our notion of weak solutions will be based on an energy inequality, which, in turn, suggests

suitable function spaces for (ρε, vε). Furthermore, we shall derive variational formulations for the partial diﬀerential equations (3.2.1) and (3.2.2). To be more accurate, throughout this section we will use the following assumptions.

Assumptions 3.2.2 (Existence of smooth solutions). In addition to Assumptions 3.2.1, for each ε> 0, let the following assumptions hold true.

(i) (i) 1 0 n 1. Prescribed initial values. (ρε , vε ) C (Ω) C (Ω) are prescribed initial values ∈ × with ρ(i) 0 in Ω and ρ(i)(x)dx β Ω , β Ω , ε ≥ ε ∈ 1 | | 2 | | ZΩ where β1 and β2 are the zeros of the double-well potential as in Assumptions 3.1.1. Additionally, the initial energy

2 2 tot,(i) 1 (i) 1 (i) 1 (i) (i) tot Eε = W (ρε )+ ε ρε + ρε vε dx = Eε (0) (3.2.7) Ω ε 2 ∇ 2 ≤E Z

is bounded by some constant > 0 independent of ε. E

2. Existence of smooth solutions. (ρε, vε) is a solution of (3.2.1)–(3.2.6) belonging to n C3,1(Ω (0, T )) C1(Ω [0, T )) C2,1(Ω (0, T )) C0(Ω [0, T )) . × ∩ × × × ∩ × 3. Non-negative density. ρε is non-negative, i.e., there holds ρε 0 in Ω (0, T ). ≥ ×

38 3.2 The Notion of Weak Solutions

3.2.1 Energy Equality

For solutions (ρε, vε) of the diﬀuse-interface model (3.2.1)–(3.2.6), we shall study the tot corresponding energy functionals Eε, E : [0, T ) R given by ε →

1 ε 2 Eε(t)= W (ρε( ,t)) + ρε( ,t) dx (3.2.8) ε · 2 |∇ · | ZΩ and tot 1 ε 2 1 2 E (t)= W (ρε( ,t)) + ρε( ,t) + ρε( ,t) vε( ,t) dx (3.2.9) ε ε · 2 |∇ · | 2 · | · | ZΩ for t [0, T ). The latter turns out to be non-increasing in t, as we now show. ∈

Theorem 3.2.3 (Energy equality). Let Assumptions 3.2.2 be satisfied. Then, for fixed ε (0, 1), the energy functional Etot, introduced in (3.2.9), satisfies the identity ∈ ε

d tot 2 E (t)= 2 µ(ρε(x,t)) Dvε(x,t) dx (3.2.10) dt ε − | | ZΩ for all t (0, T ). Moreover, for all τ ,τ [0, T ) such that τ τ , there holds ∈ 1 2 ∈ 1 ≤ 2

τ2 tot 2 tot E (τ ) + 2 µ(ρε) Dvε dx dt = E (τ ). (3.2.11) ε 2 | | ε 1 Zτ1 ZΩ tot In particular, Eε is non-increasing.

1 ε 2 1 2 Proof. Calculating the time derivative of e = e(ρε, vε)= W (ρε)+ ρε + ρε vε , we ε 2 |∇ | 2 | | get

1 ′ 1 1 ∂te = W (ρε)∂tρε + ε ρε ∂tρε + ∂t(ρεvε) vε + ρεvε ∂tvε ε ∇ ·∇ 2 · 2 · 1 ′ 1 2 = W (ρε)∂tρε + ε ρε ∂tρε + ∂t(ρεvε) vε ∂tρε vε ε ∇ ·∇ · − 2 | | 1 ′ 1 2 = W (ρε) vε ∂tρε + ε ρε ∂tρε + ∂t(ρεvε) vε. ε − 2 | | ∇ ·∇ · 1 ′ Abbreviating wε = ε∆ρε + W (ρε) and replacing all time derivatives with the help of − ε the diﬀerential equations (3.2.1) and (3.2.2) leads to

1 2 1 ′ ∂te = vε W (ρε) div(ρεvε) ε ρε div(ρεvε) div(ρεvε vε) vε 2 | | − ε − ∇ ·∇ − ⊗ · + 2 div(µ(ρε)Dvε) vε ρε wε vε. · − ∇ ·

1 2 1 2 Using the identity div(ρεvε vε) vε = div(ρεvε) vε + div(ρε vε vε) yields ⊗ · 2 | | 2 | |

1 ′ ∂te = W (ρε) div(ρεvε) ε ρε div(ρεvε)+2div(µ(ρε)Dvε) vε − ε − ∇ ·∇ · 1 2 ρε wε vε div(ρε vε vε). − ∇ · − 2 | | Let t (0, T ). As there holds ∈ d d Etot(t)= e(x,t)dx = ∂ e(x,t)dx, dt ε dt t ZΩ ZΩ

39 Chapter 3 The Diﬀuse-Interface Model

we obtain by integration by parts

d Etot(t) dt ε 1 ′ = W (ρε) div(ρεvε) ε ρε div(ρεvε)+2div(µ(ρε)Dvε) vε dx − ε − ∇ ·∇ · ZΩ 1 2 ρε wε vε + div(ρε vε vε) dx − ∇ · 2 | | ZΩ 1 2 2 = div(ρεwεvε)+ div(ρε vε vε) + 2µ(ρε) Dvε dx − 2 | | | | ZΩ n−1 n−1 ε div(ρεvε) ρε ν d (x) + 2 µ(ρε)(Dvεvε) ν d (x) − ∇ · H · H Z∂Ω Z∂Ω 2 = 2 µ(ρε) Dvε dx − | | ZΩ 1 2 n−1 + 2µ(ρε)Dvεvε ε div(ρεvε) ρε ρεwεvε ρε vε vε ν d (x). − ∇ − − 2 | | · H Z∂Ω Now, in view of the boundary conditions (3.2.3) and (3.2.4), the boundary integral vanishes and we obtain (3.2.10).

Integrating (3.2.10) over an arbitrary time interval (τ1,τ2) leads to (3.2.11).

tot τ2 2 It follows that E is non-increasing, since 2 µ(ρε) Dvε dx dt is non-negative by ε τ1 Ω | | Assumptions 3.2.1. This completes the proof. R R

3.2.2 Variational Formulations

To derive a weak formulation of (3.2.1)–(3.2.6), we provide the following variational formulations of the mass balance (3.2.1) and the balance of linear momentum (3.2.2).

Lemma 3.2.4 (Variational formulations). Assumptions 3.2.2 imply the following variational formulations.

1. Weak form of mass balance. For every ϕ C∞ (Ω [0, T )), there holds ∈ (0) ×

T (i) ρε∂tϕ + ρεvε ϕ dx dt + ρ ϕ( , 0) dx = 0. ·∇ ε · Z0 ZΩ ZΩ

2. Weak form of linear-momentum balance. For every ψ C∞ (Ω [0, T ))n, there ∈ (0) × holds

T 1 ρεvε∂tψ + ρεvε vε : ψ + p(ρε) div(ψ) 2µ(ρε)Dvε : Dψ dx dt ⊗ ∇ ε − Z0 ZΩ = ρ(i)v(i) ψ( , 0) dx (3.2.12) − ε ε · · ZΩ T 1 2 ε ρε div(ψ)+ ρε ρε div(ψ)+ ρε ρε : ψ dx dt. − 2 |∇ | ∇ ·∇ ∇ ⊗∇ ∇ Z0 ZΩ

Proof. 1. Multiplying (3.2.1) by ϕ C∞ (Ω [0, T )) and integrating with respect to space ∈ (0) ×

40 3.2 The Notion of Weak Solutions

and time leads to T 0= (∂tρε + div(ρεvε))ϕ dx dt Z0 ZΩ T = ρε∂tϕ + ρεvε ϕ dx dt ρε( , 0)ϕ( , 0) dx − ·∇ − · · Z0 ZΩ ZΩ T n−1 + (ρεvε ν)ϕ d (x)dt. · H Z0 Z∂Ω Due to the initial condition (3.2.5) and the boundary condition (3.2.4), we get

T (i) ρε∂tϕ + ρεvε ϕ dx dt + ρ ϕ( , 0) dx = 0. ·∇ ε · Z0 ZΩ ZΩ 2. Multiplying (3.2.2) by ψ C∞ (Ω [0, T ))n and integrating with respect to space and ∈ (0) × time leads to T 1 0= ∂t(ρεvε) + div(ρεvε vε)+ p(ρε) 2 div(µ(ρε)Dvε) ψ dx dt ⊗ ε ∇ − · Z0 ZΩ T ε ρε ∆ρε ψ dx dt − ∇ · Z0 ZΩ T 1 = ρεvε∂tψ + ρεvε vε : ψ + p(ρε) div(ψ)dx dt − ⊗ ∇ ε Z0 ZΩ T + 2µ(ρε)Dvε : Dψ ε ρε div(ρεψ)dx dt − ∇ ·∇ Z0 ZΩ ρε( , 0)vε( , 0) ψ( , 0) dx. − · · · · ZΩ Taking into account the initial conditions (3.2.5) and (3.2.6) implies

T 1 ρεvε∂tψ + ρεvε vε : ψ + p(ρε) div(ψ) 2µ(ρε)Dvε : Dψ dx dt ⊗ ∇ ε − Z0 ZΩ T (3.2.13) (i) (i) = ρ v ψ( , 0) dx ε ρε div(ρεψ)dx dt. − ε ε · · − ∇ ·∇ ZΩ Z0 ZΩ Integrating the identity

ρε div(ρεψ) ∇ ·∇ = ρε (ρε div(ψ)+ ρε ψ) ∇ ·∇ ∇ · 2 1 2 = ρε div(ψ)+ ρε ρε div(ψ)+ ρε ψ + ρε ρε : ψ |∇ | ∇ ·∇ 2 ∇ |∇ | · ∇ ⊗∇ ∇ over Ω (0, T ) and using integration by parts leads to × T ρε div(ρεψ)dx dt ∇ ·∇ Z0 ZΩ T 2 1 2 = ρε div(ψ)+ ρε ρε div(ψ)+ ρε ψ dx dt |∇ | ∇ ·∇ 2 ∇ |∇ | · Z0 ZΩ T + ρε ρε : ψ dx dt ∇ ⊗∇ ∇ Z0 ZΩ T 1 2 = ρε div(ψ)+ ρε ρε div(ψ)+ ρε ρε : ψ dx dt. 2 |∇ | ∇ ·∇ ∇ ⊗∇ ∇ Z0 ZΩ Together with (3.2.13), this implies (3.2.12). This completes the proof.

41 Chapter 3 The Diﬀuse-Interface Model

3.2.3 The Weak Formulation

The energy estimates and the variational formulations, derived in Theorem 3.2.3 and Lemma 3.2.4, suggest suitable function spaces and a weak formulation of the system (3.2.1)–(3.2.6). Moreover, assuming that the initial total energy Etot,(i) is uniformly bounded in ε > 0, by Theorem 3.2.3, it is convenient to consider solutions which obey an appropriate energy inequality, cf. also Remark 4.3.16 below. For ﬁxed ε > 0, (3.2.11)

also implies control on ρε. This gives rise to the following definition. ∇ Definition 3.2.5 (Weak solutions to the diffuse-interface model). Suppose that Assump- tions 3.2.1 hold true. Let ε > 0 and let p∗ > 2 be as in (3.1.1). For prescribed initial (i) (i) p∗ 1 1 n (i) values (ρε , vε ) belonging to (L (Ω) H (Ω)) H (Ω) with ρε 0 a.e. in Ω, a pair ∩ × 0 ≥ (ρε, vε) is called a weak solution to (3.2.1)–(3.2.6) if the following conditions are satisfied.

∞ 1 ∞ p∗ 1. Regularity of density. ρε L (0, T ; H (Ω)) L (0, T ; L (Ω)). ∈ ∩ 2 1 n 2. Regularity of velocity. vε L (0, T ; H (Ω) ). ∈ 0

3. Non-negative density. ρε 0 a.e. in Ω (0, T ). ≥ × 4. Weak form of mass balance. For all ϕ C∞ (Ω [0, T )), there holds ∈ (0) × T (i) ρε∂tϕ + ρεvε ϕ dx dt + ρ ϕ( , 0) dx = 0. (3.2.14) ·∇ ε · Z0 ZΩ ZΩ

5. Weak form of linear-momentum balance. For all ψ C∞ (Ω [0, T ))n, there ∈ (0) × holds

T 1 ρεvε ∂tψ + ρεvε vε : ψ + p(ρε) div(ψ) 2µ(ρε)Dvε : Dψ dx dt · ⊗ ∇ ε − Z0 ZΩ = ρ(i)v(i) ψ( , 0) dx (3.2.15) − ε ε · · ZΩ T 1 2 ε ρε ρε : ψ + ρε div(ψ)+ ρε ρε div(ψ)dx dt. − ∇ ⊗∇ ∇ 2 |∇ | ∇ ·∇ Z0 ZΩ

6. Energy inequality. Let Etot be as in (3.2.9). For a.e. τ [0, T ), including τ = 0, ε 1 ∈ 1 the following estimate holds

τ2 tot 2 tot E (τ ) + 2 µ(ρε) Dvε dx dt E (τ ) (3.2.16) ε 2 | | ≤ ε 1 Zτ1 ZΩ for all τ [τ , T ). 2 ∈ 1 From now on, we will always consider solutions in the sense of the foregoing deﬁnition.

Remark 3.2.6 (Lower semi-continuity of the energy functional). The energy functional

(ρ, v) Etot(ρ, v)= 1 W (ρ)+ ε ρ 2 + 1 ρ v 2 dx 7→ ε ε 2 |∇ | 2 | | ZΩ

42 3.3 A Priori Estimates is lower semi-continuous on the “natural” function space H1(Ω) L2(Ω)n. Moreover, by × the energy estimate (3.2.16), the function t Etot(t) is monotonically non-increasing in 7→ ε a.e. t [0, T ). Hence, by possibly changing the values of (ρε(t), vε(t)) on a null set with ∈ respect to time, we can always achieve that t Etot(t) is lower semi-continuous on [0, T ); 7→ ε that is, for any sequence (tm)m N [0, T ) and any t [0, T ) such that tm t as m , ∈ ⊂ ∈ → →∞ we have tot tot E (t) lim inf E (tm). ε ≤ m→∞ ε tot Hence there is no loss of generality in assuming that Eε is lower semi-continuous. This, in turn, justifies to require (3.2.16) for all τ [τ , T ) instead of merely for a.e. τ [τ , T ). 2 ∈ 1 2 ∈ 1 Remark 3.2.7 (Existence of weak solutions). Throughout this work we will always assume the existence of weak solutions to the system (3.2.1)–(3.2.6) in the sense of the above definition. To the best of our knowledge, there is no global-existence result known. The question of short-time existence has been addressed by Kotschote [61, 62]. These results ensure the (short-time) existence of strong solutions, i.e., the functions ρ and v and all their derivatives appearing in the partial differential equations (3.2.1) and (3.2.2) belong, respectively, to the spaces Lq(0, T ; Lq(Ω)) and Lq(0, T ; Lq(Ω)n) for some q (1, ). This, ∈ ∞ in particular, guarantees the existence of weak solutions in sense of the foregoing definition for short times. (Note that the interval of existence may depend on ε, in general.) The existence of global weak solutions in the case of a (non-physical) monotone pressure function has been investigated by Haspot [49].

3.3 A Priori Estimates

We are interested in a priori estimates for families of weak solutions (ρε, vε)ε∈(0,1) to the diffuse-interface model (3.2.1)–(3.2.6) in the sense of Definition 3.2.5. Deriving uniform bounds on (ρε, vε)ε∈(0,1) is a first step towards compactness results. To this end, we will basically proceed in three steps. By definition, the class of weak solutions is restricted by an energy constraint, and we shall show that this already leads to a priori estimates. The bound on (ρε)ε∈(0,1) needs improving, which we do in the remainder of this section. We will derive uniform bounds with respect to the spatial variables in the second step, and, lastly, we will improve the time regularity.

3.3.1 A Priori Estimates arising from Energy Considerations

We shall study weak solutions (ρε, vε)ε∈(0,1) of the Navier–Stokes–Korteweg model (3.2.1)– (3.2.6) with respect to well-prepared initial data.

Assumptions 3.3.1 (Existence of weak solutions). Suppose that Assumptions 3.2.1 hold true. Let (ρε, vε)ε∈(0,1) be a family of weak solutions of (3.2.1)–(3.2.6) in the sense of

43 Chapter 3 The Diﬀuse-Interface Model

Deﬁnition 3.2.5, with uniformly bounded initial energy. That is, for the initial data (i) (i) (ρε , vε ) , there exists a constant > 0, which does not depend on ε, such that, ε∈(0,1) E tot,(i) tot,(i) for every ε (0, 1), there holds Eε , where Eε is given by (3.2.7). ∈ ≤E

The ﬁrst a priori estimates for weak solutions (ρε, vε)ε>0 can be obtained by combining energy equality (3.2.16) with the growth properties of the double–well potential derived in Lemma 3.1.4.

Theorem 3.3.2 (A priori estimates). If Assumptions 3.3.1 hold true, then there exists a constant C > 0, independent of ε (0, 1), such that, for p > 2 as in (3.1.1), a = β1+β2 ∈ ∗ 2 β2−β1 and b = 2 , there holds

ρε ∞ p∗ + vε 2 1 n C (3.3.1) k kL (0,T ;L (Ω)) k kL (0,T ;H (Ω) ) ≤ and

ρε a b ∞ 2 C√ε. (3.3.2) k | − |− kL (0,T ;L (Ω)) ≤ Proof. Assumptions 3.2.1 and the energy equality (3.2.16), for t (0, T ), imply that ∈ t t 2 2 tot,(i) tot 2cµ Dvε dx dt 2 µ(ρε) Dvε dx dt = E E (t). | | ≤ | | ε − ε Z0 ZΩ Z0 ZΩ tot As Eε is non-negative, by the monotone-convergence theorem, we see that

T 2 tot,(i) 2cµ Dvε dx dt E . | | ≤ ε Z0 ZΩ Hence, by Korn’s inequality [84, Theorem 1.33], there exists some constant C > 0 such that T 2 tot,(i) vε dx dt CE . |∇ | ≤ ε Z0 ZΩ 2 1 n tot,(i) Thus, as vε L (0, T ; H (Ω) ) and as the initial energy Eε is uniformly bounded by ∈ 0 Assumptions 3.3.1, we obtain the bound for (vε)ε∈(0,1).

For the proof of the bounds on (ρε) , we recall that ρε 0 a.e. in Ω (0, T ) by ε∈(0,1) ≥ × Deﬁnition 3.2.5. In view of the energy inequality (3.2.16), for ε (0, 1) and a.e. t (0, T ), ∈ ∈ we infer that 1 tot tot,(i) W (ρε(t)) dx E (t) E . ε ≤ ε ≤ ε ≤E ZΩ Finally, for a.e. t (0, T ), Lemma 3.1.4 yields ∈

p∗ C ρε(t) a dx W (ρε(t)) dx + C Ω ε + C Ω , 1 | − | ≤ 2 | |≤ E 2 | | ZΩ ZΩ

which implies the bound for (ρε)ε∈(0,1) in (3.3.1), and

2 ( ρε(t) a b) dx C W (ρε(t)) dx C ε . | − |− ≤ 1 ≤ 1 E ZΩ ZΩ This ﬁnishes the proof of the theorem.

44 3.3 A Priori Estimates

The a priori estimate (3.3.1) directly allows one to extract a weakly converging subsequence 2 1 n of (vε)ε∈(0,1) due to the reﬂexivity of the space L (0, T ; H (Ω) ) (see Theorem 3.4.1 below), whereas extracting convergent subsequences of (ρε)ε∈(0,1) requires more work. Following ideas of Chen [29], we will derive further bounds for (ρε)ε∈(0,1). This improvement will be done separately with respect to space and time variables.

3.3.2 The Modica–Mortola Trick and BV -Bounds

We shall study the family ( ρε) , which, by means of the energy estimate (3.2.16), ∇ ε∈(0,1) ∞ 2 n is a sequence in L (0, T ; L (Ω)) . Nevertheless, ( ρε) is not uniformly bounded in ∇ ε∈(0,1) L∞(0, T ; L2(Ω))n. To circumvent this problem, we will use the transformation

s Φ(s)= min 1 W (z), z a 2 + b2 dz for s [0, ) (3.3.3) 2 | − | ∈ ∞ Za q and will derive a gradient bound for the transformed sequence (rε)ε∈(0,1) deﬁned by

rε =Φ ρε. (3.3.4) ◦

The idea to study rε, instead of ρε, goes back to Modica and Mortola [73, 74], cf. also [29]. To justify its introduction, ﬁrstly, we will study the transformation Φ. In the following lemma, we will transfer the growth conditions on the double-well potential W , obtained in Lemma 3.1.4, to Φ and will show that Φ is bijective onto its image. Secondly, we will apply the so-called Modica–Mortola trick (see (3.3.13) below) to show that (rε(t))ε∈(0,1) is uniformly bounded in BV (Ω).

Lemma 3.3.3 (Growth of Φ). Let Φ: [0, ) R be deﬁned as in (3.3.3) and suppose that ∞ → Assumptions 3.1.1 are satisﬁed. Then Φ C1([0, )), and there exist constants C ,C > 0 ∈ ∞ 1 2 such that

C s s 2 Φ(s ) Φ(s ) C s s ( s a + s a + b) (3.3.5) 1 | 2 − 1| ≤ | 2 − 1 |≤ 2 | 2 − 1| | 1 − | | 2 − | for all s ,s [0, ). In particular, Φ is bijective onto its image, which is [Φ(0), ). 1 2 ∈ ∞ ∞

Proof. We split the proof into several steps. Step 1. The fundamental theorem of calculus implies that Φ C1([0, )) because of the ∈ ∞ continuity of the integrand

z min 1 W (z), z a 2 + b2 . 7→ 2 | − | q Step 2. For the proof of (3.3.5), we may, without loss of generality, assume that s2 > s 0. Lemma 3.1.4 ensures the existence of some constant C > 0 such that W (z) 1 ≥ ≥ C ( z a b)2 for all z [0, ). Since, for all z [0, ), there holds z a 2 + b2 | − |− ∈ ∞ ∈ ∞ | − | ≥

45 Chapter 3 The Diﬀuse-Interface Model

( z a b)2, abbreviating M = min 1 C, 1 , we see that | − |− { 2 } s2 1 2 2 Φ(s2) Φ(s1)= min 2 W (z), z a + b dz − s | − | Z 1 q s2 √M z a b dz, ≥ || − |− | Zs1 s2−a √M z b dz. ≥ || |− | Zs1−a To prove the ﬁrst inequality in (3.3.5), for z , z R with z < z , we shall verify that 1 2 ∈ 1 2 z2 z b dz 1 z z 2 . (3.3.6) || |− | ≥ 8 | 2 − 1| Zz1 For the proof of (3.3.6), we distinguish the following ﬁve cases:

(i) 0 z < z b. ≤ 1 2 ≤ (ii) b z < z . ≤ 1 2 (iii) 0 z b < z . ≤ 1 ≤ 2 (iv) z < z 0. 1 2 ≤

(v) z1 < 0 < z2.

Case (i). As 0 z < z b, we compute ≤ 1 2 ≤ z2 z2 1 2 2 z b dz = z b dz = 2 (z1 b) (z2 b) . z || |− | − z − − − − Z 1 Z 1 This yields

z2 z b dz = 1 (z z )2 + (z z )(z b) 1 (z z )2. || |− | 2 1 − 2 1 − 2 2 − ≥ 2 1 − 2 Zz1 Case (ii). For b z < z , we proceed analogously to case (i) to get ≤ 1 2 z2 z2 1 2 2 z b dz = z b dz = 2 (z2 b) (z1 b) . z || |− | z − − − − Z 1 Z 1 Hence we infer

z2 z b dz = 1 (z z )2 + (z z )(z b) 1 (z z )2. || |− | 2 2 − 1 2 − 1 1 − ≥ 2 2 − 1 Zz1 Case (iii). For 0 z b < z , we consider the decomposition ≤ 1 ≤ 2 z2 b z2 z b dz = z b dz + z b dz. || |− | || |− | || |− | Zz1 Zz1 Zb Application of the results of the cases (i) and (ii) leads to

z2 z b dz 1 (b z )2 + 1 (z b)2 1 (z z )2. || |− | ≥ 2 − 1 2 2 − ≥ 4 2 − 1 Zz1 Case (iv). Let z < z 0. In view of the foregoing cases (i)–(iii), we obtain 1 2 ≤ −z1 z2 1 (z z )2 z b dz = z b dz, 4 2 − 1 ≤ || |− | || |− | Z−z2 Zz1

46 3.3 A Priori Estimates where the last equality holds true as the integrand z z b is symmetric about z = 0. 7→ || |− | Case (v). In the remaining case z1 < 0 < z2, we use the decomposition

z2 0 z2 z b dz = z b dz + z b dz. || |− | || |− | || |− | Zz1 Zz1 Z0 Taking into account the foregoing cases, we therefore obtain

z2 z b dz 1 z2 + 1 z2 1 (z z )2. || |− | ≥ 4 1 4 2 ≥ 8 2 − 1 Zz1 This concludes the proof of the ﬁrst estimate in (3.3.5). Step 3. The second estimate in (3.3.5) can be obtained as follows. Since, for all z (s ,s ), ∈ 1 2 there holds z a 2 + b2 z a + b s a + s a + b, | − | ≤ | − | ≤ | 1 − | | 2 − | q we conclude that

s2 Φ(s ) Φ(s ) = min 1 W (z), z a 2 + b2 dz | 2 − 1 | 2 | − | Zs1 q s s ( s a + s a + b). ≤ | 2 − 1| | 1 − | | 2 − | Step 4. As a direct consequence of its deﬁnition, Φ is strictly monotonically increasing on [0, ). Hence Φ is bijective onto its image Φ([0, )) = [Φ(0), ). ∞ ∞ ∞

We are interested in the eﬀect of the transformation Φ, introduced in (3.3.3), on the sequence (ρε)ε∈(0,1). For technical reasons, we extend Φ to the entire real line to a function Φ: R R by → e Φ(s) if s 0, Φ(s)=  ≥ (3.3.7)  Φ( s) + 2Φ(0) if s 0. e − − ≤ In the sequel, for the sake of convenience, we will not distinguish between the transformation Φ: [0, ) R, its extension Φ: R R and the corresponding Nemickii operator ∞ → → e u Φ˜ u. (3.3.8) 7→ ◦

We shall simply denote any of these mappings by Φ, wherever no confusion can arise. In order to explore the continuity of the above Nemickii operator between Lebesgue and Sobolev spaces, we need to estimate the growth of Φ and its derivative on the real line.

Proposition 3.3.4 (Growth of extension). Let Assumption 3.1.1 hold true. The extension Φ: R R, introduced in (3.3.7), is bijective onto R and belongs to C1(R). Furthermore, → there exist constants D , D > 0 such that, for any s R, there holds 1 2 ∈

Φ(s) D s 2 + D and Φ′(s) D s + D . (3.3.9) | |≤ 1 | | 2 ≤ 1 | | 2

47 Chapter 3 The Diﬀuse-Interface Model

Proof. In view of Lemma 3.3.3 and by construction of the extension (3.3.7), Φ is bijective and belongs to C1(R). Due to (3.3.7), for any s< 0, we have

Φ(s) 2 Φ(0) + Φ( s) and Φ′(s) = Φ′( s) . | |≤ | | | − | −

Therefore, without loss of generality, in the remainder of this proof let s 0. ≥ In view of the growth estimate (3.3.5), there holds

Φ(s) = Φ(s) Φ(a) C s a 2 + C b s a . | | | − |≤ 2 | − | 2 | − |

With the help of Young’s inequality and by possibly adjusting the constants, we infer the ﬁrst estimate in (3.3.9). Furthermore, by (3.3.3), there holds

Φ′(s) = min 1 W (s), s a 2 + b2 s a 2 + b2 s a + b, 2 | − | ≤ | − | ≤ | − | q q which implies the estimate for Φ′ in (3.3.9). This completes the proof.

From the growth estimates for Φ, we now conclude to the continuity of the corresponding Nemickii operators.

Proposition 3.3.5 (Nemickii operators). Let Assumptions 3.2.1 be satisﬁed. The function Φ, deﬁned in (3.3.7), induces, by (3.3.8), continuous and bounded operators

Φ: L2(Ω) L1(Ω) and Φ: H1(Ω) W 1,1(Ω). → → Moreover, for every u H1(Ω), there holds Φ u W 1,1(Ω) and ∈ ◦ ∈ (Φ u)=(Φ′ u) u. (3.3.10) ∇ ◦ ◦ ∇

1 ∞ 1 Proof. Let u H (Ω). Then there exists a sequence (uk)k N C (Ω) H (Ω) with ∈ ∈ ⊂ ∩ 1 ∞ n uk u in H (Ω) for k . For any ψ C (Ω) and any k N, we then have → →∞ ∈ 0 ∈

′ Φ(uk) div(ψ)dx = (Φ(uk)) ψ dx = Φ (uk) uk ψ dx. (3.3.11) − ∇ · ∇ · ZΩ ZΩ ZΩ The growth condition (3.3.9) guarantees that the Nemickii operators

Φ: L2(Ω) L1(Ω), u Φ u and Φ′ : L2(Ω) L2(Ω), u Φ′ u → 7→ ◦ → 7→ ◦ are continuous and bounded, cf. [86, Section 3.1.2]. Consequently, for k , there holds →∞ 1 ′ ′ 1 n Φ(uk) Φ(u) in L (Ω) and Φ (uk) uk Φ (u) u in L (Ω) . Letting k in (3.3.11), → ∇ → ∇ →∞ therefore yields Φ(u) div(ψ)dx = Φ′(u) u ψ dx, − ∇ · ZΩ ZΩ which proves (3.3.10). Finally, we may use (3.3.10) to estimate

′ ′ (Φ(u)) 1 n = Φ (u) u 1 n Φ (u) 2 u 2 n . k∇ kL (Ω) k ∇ kL (Ω) ≤ k kL (Ω)k∇ kL (Ω)

48 3.3 A Priori Estimates

This implies

2 2 2 Φ(u) 1,1 = Φ(u) 1 + (Φ(u)) 1 n k kW (Ω) k kL (Ω) k∇ kL (Ω) 2 ′ 2 2 Φ(u) 1 + Φ (u) 2 u 2 n . ≤ k kL (Ω) k kL (Ω)k∇ kL (Ω) Now the continuity and boundedness of the Nemickii operators Φ: L2(Ω) L1(Ω) and → Φ′ : L2(Ω) L2(Ω) imply that Φ: H1(Ω) W 1,1(Ω) is continuous and bounded. This → → ﬁnishes the proof.

∞ 2 Combining the growth estimate (3.3.5) and the L (0, T ; L (Ω))-bound (3.3.2) for ρε, we are able to prove uniform bounds on rε. This justifies the definition of rε. ∇ Lemma 3.3.6 (Uniform BV -bounds). Suppose that Assumptions 3.3.1 are satisfied and let rε be as in (3.3.4). Then there exists a constant C > 0, independent of ε, such that

rε ∞ 1,1 C (3.3.12) k kL (0,T ;W (Ω)) ≤

∞ for all ε (0, 1). In particular, (rε) is uniformly bounded in L (0, T ; BV (Ω)). ∈ ε∈(0,1)

Proof. By the a priori estimate (3.3.1), (ρε)ε>0 is uniformly bounded in

.((L∞(0, T ; Lp∗ (Ω)) ֒ L∞(0, T ; L2(Ω →

In view of Proposition 3.3.5, we conclude that rε = Φ ρε is uniformly bounded in ◦ ∞ 1 ∞ 1 L (0, T ; L (Ω)). For ﬁxed ε (0, 1), there holds ρε L (0, T ; H (Ω)) by Deﬁnition 3.2.5. ∈ ∈ ∞ 1,1 Again by Proposition 3.3.5 and using (3.3.3), we deduce that rε L (0, T ; W (Ω)) and ∈

′ 1 2 2 rε =Φ (ρε) ρε = min W (ρε), ρε a + b ρε. ∇ ∇ 2 | − | ∇ q By Young’s inequality, it follows that

1 1 1 1 2 rε W (ρε) ρε W (ρε)+ ε ρε . |∇ |≤ 2 |∇ |≤ 2 ε 2 |∇ | q Due to (3.2.16) and Assumptions 3.3.1, for a.e. t (0, T ), we conclude that ∈

1 1 2 tot 2 rε(x,t) dx W (ρε(x,t)) + ε ρε(x,t) dx E (t) , (3.3.13) |∇ | ≤ ε 2 |∇ | ≤ ε ≤E ZΩ ZΩ tot where Eε is given by (3.2.9). This implies (3.3.12). Remarking that the embedding .W 1,1(Ω) ֒ BV (Ω) is continuous, see Remark 2.3.4, concludes the proof →

3.3.3 H¨older Bounds with Respect to Time

The a priori bound on ρε, proved in Theorem 3.3.2, allows us to extract weakly converging p∗ subsequences in L (Ω) of (ρε(t))ε>0. However, the choice of the subsequence may diﬀer for each t (0, T ). To overcome this drawback, we will study the H¨older continuity of ∈ the functions ρε with respect to the time variable. Adapting ideas of Chen [29] to our

49 Chapter 3 The Diﬀuse-Interface Model

situation, we will establish uniform H¨older bounds on (ρε)ε∈(0,1). This will ensure the existence of a convergent subsequence independent of t. To state our result, we recall that, for a Banach space Y , the space C0([0, T ]; Y ) contains all continuous functions f : [0, T ] Y and is equipped with the norm →

f C0([0,T ];Y ) = sup f(t) Y . k k t∈[0,T ] k k

For α (0, 1), the H¨older space C0,α([0, T ]; Y ) consists of all functions f C0([0, T ]; Y ) ∈ ∈ that additionally satisfy

f(t2) f(t1) Y f C0,α([0,T ];Y ) = f C0([0,T ];Y ) + sup k − α k < . k k k k t

functions ρε and rε.

Proposition 3.3.7 (Time continuity). Suppose that Assumptions 3.3.1 are satisﬁed and 0 2 0 1 let rε be given by (3.3.4). Then ρε and rε belong to C ([0, T ]; L (Ω)) and C ([0, T ]; L (Ω)), respectively. Moreover, there exists a constant C > 0, independent of ε, such that, for all ε (0, 1), there holds ∈

ρε 0 2 + rε 0 1 C. (3.3.14) k kC ([0,T ];L (Ω)) k kC ([0,T ];L (Ω)) ≤

Proof. As a consequence of Theorem 3.3.2 and Lemma 3.3.6, ρε and rε are uniformly bounded in L∞(0, T ; L2(Ω)) and L∞(0, T ; L1(Ω)), respectively. What is left is to show 2 1 that, for ε (0, 1), ρε and rε are continuous functions taking values in L (Ω) and L (Ω), ∈ respectively. To this end, let ε (0, 1) be ﬁxed in the remainder of this proof. ∈ In view of Assumptions 3.2.1, we consider dimension n = 2 or dimension n = 3, where the embedding H1(Ω) ֒ L6(Ω) is continuous. Therefore, by Deﬁnition 3.2.5, we have → ∞ 1 ∞ 6 ((ρε L (0, T ; H (Ω)) ֒ L (0, T ; L (Ω ∈ → and 2 1 n 2 6 n .( (vε L (0, T ; H (Ω) ) ֒ L (0, T ; L (Ω ∈ 0 →

In view of (3.2.14), the distributional time derivative ∂tρε satisﬁes

2 3 ∂tρε = div(ρεvε)= ρε div(vε) ρε vε L (0, T ; L 2 (Ω)). − − −∇ · ∈

Altogether, we infer that ρε belongs to

∞ 1 1,2 3 2 1 1,2 −1 ,((L (0, T ; H (Ω)) W (0, T ; L 2 (Ω)) ֒ L (0, T ; H (Ω)) W (0, T ; H (Ω ∩ → ∩ such that the continuity of the embedding

(L2(0, T ; H1(Ω)) W 1,2(0, T ; H−1(Ω)) ֒ C0([0, T ]; L2(Ω)), (3.3.15 ∩ →

50 3.3 A Priori Estimates

0 2 see [15, Chapter III, Theorem 4.10.2] and [67, Th´eor`eme 12.4], yields ρε C ([0, T ]; L (Ω)). ∈ 2 1 Finally, we may use that rε =Φ ρε, by (3.3.4), and that Φ: L (Ω) L (Ω) is a continuous ◦ → and bounded operator, see Proposition 3.3.5, which establishes (3.3.14).

2 1 Remark 3.3.8. Due to the previous proposition, ρε(t) L (Ω) and rε(t) L (Ω) are ∈ ∈ well-deﬁned for any t [0, T ]. ∈

3.3.3.1 Extension of Density Function

For technical reasons, we shall mollify the density function ρε. To this end, for some small n parameter η > 0, we extend ρε to the set x R : dist(x, Ω) η [0, T ] in a suitable 0 { ∈ ≤ 0}× n way. For η > 0, let Tη = x R : dist(x, ∂Ω) η denote the η-tube of ∂Ω, and deﬁne { ∈ ≤ } n the signed distance function d∂ : R R to ∂Ω by Ω → dist(x, ∂Ω), for x Rn Ω,  ∈ \ d∂Ω(x)= 0, for x ∂Ω,  ∈  dist(x, ∂Ω), for x Ω. − ∈  The boundary ∂Ω of Ω is of classC2 by Assumptions 3.2.1. Therefore, there exists some 2 η > 0 such that d∂ C (Tη ), and every x Tη has a unique representation 0 Ω ∈ 0 ∈ 0

x = s + ην(s)= ∂ (x)+d∂ (x)ν( ∂ (x)), P Ω Ω P Ω where s = ∂ (x) ∂Ω is the projection of x onto ∂Ω and η [ η , η ] equals d∂ (x). In P Ω ∈ ∈ − 0 0 Ω the remainder of this section we ﬁx this choice of η0. Moreover, the mapping

Π: Tη ∂Ω [ η , η ], x ( ∂ (x), d∂ (x)) 0 → × − 0 0 7→ P Ω Ω

1 n+1 is bijective and belongs to C (Tη0 ) . For details and proofs we refer to [52, Section 4.6].

Abbreviating Ωη = Ω Tη , we now introduce the extensionsρ ˜ε, r˜ε :Ωη [0, T ] R of 0 ∪ 0 0 × → ρε and rε given by

ρε(x,t), for x Ω [0, T ], ρ˜ε(x,t)=  ∈ × ρε(s ην(s),t), for x = s + ην(s) Tη Ω, t [0, T ]  − ∈ 0 \ ∈  andr ˜ε = Φ ρ˜ε, respectively. The extensionρ ˜ε is constructed by “reﬂection at the ◦ boundary”. This is a common technique for extending Sobolev functions, cf. for example

[8, Chapter 5]. As a consequence, the bounds on ρε and rε carry over to their respective extensions, i.e., for some constant C > 0, independent of ε and η, we have

ρ˜ε C0([0,T ];L2(Ω )) + r˜ε C0([0,T ];L1(Ω )) + r˜ε L∞([0,T ];W 1,1(Ω )) C. k k η0 k k η0 k k η0 ≤

For convenience, in the sequel we will omit the symbol ˜ and we will write ρε and rε instead · ofρ ˜ε andr ˜ε.

51 Chapter 3 The Diﬀuse-Interface Model

Remark 3.3.9 (Boundary regularity). For the above construction of the extensions of 2 the functions ρε and rε, we essentially used the C -regularity of the boundary ∂Ω to make them well-deﬁned. In fact, all other results of the present chapter are valid under the weaker assumption that ∂Ω is merely Lipschitz.

3.3.3.2 Molliﬁcation of Density Function

We aim to approximate ρε(t) by mollification, cf. [8, Theorem 2.29] or [29, Lemma 3.2]. ∞ n n To this end, we choose a function Θ C (R ) with 0 Θ 1 on R , Rn Θdx = 1 and ∈ 0 ≤ ≤ supp(Θ) B (0). For ε (0, 1) and η (0, η ], define the mollification ρε,ηR :Ω [0, T ] R ⊂ 1 ∈ ∈ 0 × → by

ρε,η(x,t)= Θ(y)ρε(x ηy,t)dy for (x,t) Ω [0, T ]. (3.3.16) − ∈ × ZB1(0) From the deﬁnition of ρε,η, we can estimate the dependence of ρε,η on the parameter η ∇ and control the limiting behaviour of ρε,η as η tends to 0, as we now show.

Lemma 3.3.10 (Molliﬁcation of density function). Suppose that Assumptions 3.3.1 hold true. For ε (0, 1) and η (0, η ], let ρε,η be as in (3.3.16). Then ρε,η(t) belongs to ∈ ∈ 0 C∞(Rn) for all t [0, T ]. Furthermore, there exists a constant C > 0 such that, for any ∈ ε (0, 1) and all η (0, η ], the following estimates are valid. ∈ ∈ 0

1. sup ρε,η(t) ρε(t) 2 C√η. t∈[0,T ] k − kL (Ω) ≤ −( 2 n+1) 2. sup ρε,η(t) 3 n Cη 3 . t∈[0,T ] k∇ kL (Ω) ≤ Proof. Throughout this proof let ε (0, 1), η (0, η ] and t,τ [0, T ] be ﬁxed. By stand- ∈ ∈ 0 ∈ ∞ n ard properties of the convolution product [8, Theorem 2.29], ρε,η(t) belongs to C (R ). Moreover, there exists a constant C > 0, independent of ε, η, t and τ, such that

ρε,η(t) L2(Ω) ρε(t) L2(Ω ) C ρε(t) L2(Ω) k k ≤ k k η0 ≤ k k

and

ρε,η(t) ρε,η(τ) L2(Ω) ρε(t) ρε(τ) L2(Ω ) C ρε(t) ρε(τ) L2(Ω). k − k ≤ k − k η0 ≤ k − k 0 2 Hence, by Proposition 3.3.7, we see that ρε,η C ([0, T ]; L (Ω)) and, by possibly adjusting ∈ the constant C,

ρε,η 0 2 C. k kC ([0,T ];L (Ω)) ≤

1. Recall that Θdx = 1. Then, from the deﬁnition of ρ , it follows that B1(0) ε,η

R 2 2 ρε,η(t) ρε(t) dx = Θ(y)ρε(x ηy,t)dy ρε(x,t) dx Ω | − | Ω B1(0) − ! − Z Z Z 2

Θ(y) ρε(x ηy,t) ρε(x,t) dy dx. ≤ Ω B1(0) − − Z Z

52 3.3 A Priori Estimates

Application of H¨older’s inequality to the inner integral gives

2 ρε,η(t) ρε(t) 2 k − kL (Ω) 2 Θ(y) ρε(x ηy,t) ρε(x,t) dy Θ(y)dy dx ≤ | − − | ZΩ ZB1(0) ! ZB1(0) ! 2 = Θ(y) ρε(x ηy,t) ρε(x,t) dy dx. | − − | ZΩZB1(0) Recalling the deﬁnition of rε from (3.3.4) and using the growth estimate (3.3.5), the integral on the right-hand side can be estimated by

1 Θ(y) rε(x ηy,t) rε(x,t) dy dx C1 | − − | ZΩZB1(0) 1 1 = Θ(y) rε(x ξηy,t) ηy dξ dy dx C1 ∇ − · ZΩZB1(0) Z0

1 Cη rε(x ξηy,t) dξ dy dx ≤ |∇ − | ZΩZB1(0)Z0 for some constant C > 0. To further estimate the last integral, we apply Fubini’s theorem to change the order of integration, and ﬁrstly integrate with respect to x.

Using the transformation z = x ξηy and the bound on rε, obtained in Lemma 3.3.6, − ∇ results in

rε(x ξηy,t) dx rε(z,t) dz C rε(z,t) dz. Ω |∇ − | ≤ Ω |∇ | ≤ Ω |∇ | Z Z η0 Z Since the integrand on the right-hand side is independent of y and ξ, we obtain

2 ρε,η(t) ρε(t) 2 Cη rε ∞ 1,1 Cη, k − kL (Ω) ≤ k kL (0,T ;W (Ω)) ≤ 0 2 in the ﬁrst instance for a.e. t (0, T ), but as ρε,η, ρε C ([0, T ]; L (Ω)), we can go ∈ ∈ further and infer that 2 sup ρε,η(t) ρε(t) L2(Ω) Cη. t∈[0,T ] k − k ≤

2. Without loss of generality, we may assume that ρε(t) is extended by 0 outside of Ωη0 .

By the deﬁnition of ρε,η and the transformation formula, we have

ρε,η(t)= Θ(y)ρε( ηy,t)dy Rn ·− Z −n 1 = η (Θ η )( y)ρε(y,t)dy Rn ◦ ·− Z −n 1 = η (Θ ) ρε(t) , ◦ η ∗ where Θ 1 denotes the function x Θ( 1 x) and is the convolution product. By ◦ η 7→ η ∗ Young’s inequality for convolution [8, Corollary 2.25], we infer that

−n 1 ρε,η(t) 3 n = η ((Θ ) ρε(t)) 3 Rn n k∇ kL (Ω) k∇ ◦ η ∗ kL ( ) −n 1 η (Θ ) L3(Rn)n ρε(t) L1(Ω ) ≤ k∇ ◦ η k k k η0 − −n n 3 η η 3 Θ L3(Rn)n ρε(t) L1(Ω ) ≤ · k∇ k k k η0 −( 2 n+1) Cη 3 ρε(t) 1 . ≤ k kL (Ω) Proposition 3.3.7 implies the gradient estimate.

53 Chapter 3 The Diﬀuse-Interface Model

This concludes the proof.

Combining Proposition 3.3.7 and Lemma 3.3.10 leads to the desired H¨older bounds.

Theorem 3.3.11 (H¨older bounds). Let Assumptions 3.3.1 hold true. Then there exists a constant C > 0, independent of ε (0, 1), such that ∈

1 1 ρε 0, + rε 0, C. k kC 28 ([0,T ];L2(Ω)) k kC 28 ([0,T ];L1(Ω)) ≤ Proof. Let ε (0, 1). In Proposition 3.3.7, we already ensured the existence of a constant ∈ C > 0 such that

ρε 0 2 + rε 0 1 C. (3.3.17) k kC ([0,T ];L (Ω)) k kC ([0,T ];L (Ω)) ≤ In the remainder we will show that there exists a constant C > 0, independent of ε, such that 1 2 ρε(t ) ρε(t ) dx C t t 14 (3.3.18) | 2 − 1 | ≤ | 2 − 1| ZΩ and 1 rε(t ) rε(t ) dx C t t 28 (3.3.19) | 2 − 1 | ≤ | 2 − 1| ZΩ for every t ,t [0, T ]. We divide this proof into several steps. 1 2 ∈ Step 1. By Theorem 3.3.2 and as the embedding H1(Ω) ֒ L6(Ω) is continuous in → dimension n = 2, 3, the sequences (ρε)ε∈(0,1) and (vε)ε∈(0,1) are uniformly bounded in ∞ 2 2 6 n L (0, T ; L (Ω)) and L (0, T ; L (Ω) ), respectively. As (ρε, vε) is a weak solution of the diﬀuse-interface model (3.2.1)–(3.2.6), from (3.2.14), we see T T ρε∂tϕ dx dt = ρεvε ϕ dx dt 0 Ω 0 Ω ·∇ Z Z Z Z

ρε ∞ 2 vε 2 6 n ϕ 2 3 n ≤ k kL (0,T ;L (Ω))k kL (0,T ;L (Ω) )k∇ kL (0,T ;L (Ω) ) ∞ for all ϕ C (Ω (0, T )). Thus the distributional time derivative ∂tρε is uniformly ∈ (0) × 2 1,3 ∗ 1,2 1,3 ∗ bounded in L (0, T ; W (Ω) ). Hence ρε is uniformly bounded in W (0, T ; W (Ω) ). This implies, for all t ,t [0, T ] with t

(ρε(t ) ρε(t ))(ρε,η(t ) ρε,η(t )) dx 2 − 1 2 − 1 ZΩ

= ρε(t ) ρε(t ), ρε,η(t ) ρε,η(t ) , 2 − 1 2 − 1 W 1 3(Ω) t2 = div(ρεvε)(t), ρε,η(t ) ρε,η(t ) , dt − 2 − 1 W 1 3(Ω) Zt1 t2 = (ρεvε)(t) ( ρε,η(t ) ρε,η(t )) dx dt. · ∇ 2 −∇ 1 Zt1 ZΩ

54 3.3 A Priori Estimates

Now, using the estimates for ρε,η (see Lemma 3.3.10) and recalling that ρε and vε are uniformly bounded in L∞(0, T ; L2(Ω)) and L2(0, T ; L6(Ω)n), respectively, yields

(ρε(t ) ρε(t ))(ρε,η(t ) ρε,η(t )) dx 2 − 1 2 − 1 ZΩ ρεvε 3 ρε,η(t ) ρε,η(t ) 2 3 n 2 n 2 1 L (t1,t2;L (Ω) ) ≤ k kL (t1,t2;L 2 (Ω) )k∇ −∇ k

1 ρεvε 3 2 t t 2 sup ρε,η(t) 3 n 2 n 2 1 L (Ω) ≤ k kL (0,T ;L 2 (Ω) ) | − | t∈[0,T ] k∇ k ! 1 2 ρε L∞(0,T ;L2(Ω)) vε L2(0,T ;L6(Ω)n) sup ρε,η(t) L3(Ω)n t2 t1 2 ≤ k k k k t∈[0,T ] k∇ k | − | −( 2 n+1) 1 Cη 3 t t 2 . ≤ | 2 − 1|

Secondly, we may write ρε(t ) ρε(t ) = ρε(t ) ρε,η(t ) ρε,η(t ) ρε(t ) to conclude, 2 − 1 2 ∓ 2 ∓ 1 − 1 by H¨older’s inequality, that

2 ρε(t ) ρε(t ) dx | 2 − 1 | ZΩ ρε(t ) ρε(t ) 2 ρε(t ) ρε,η(t ) 2 + ρε(t ) ρε,η(t ) 2 ≤ k 2 − 1 kL (Ω) k 2 − 2 kL (Ω) k 1 − 1 kL (Ω) + (ρε(t ) ρε(t ))(ρε,η(t ) ρε,η(t )) dx. 2 − 1 2 − 1 ZΩ In view of the last two estimates, (3.3.17) and Lemma 3.3.10, there exists a constant C > 0 such that 2 −( 2 n+1) 1 ρε(t2) ρε(t1) dx C √η + η 3 t2 t1 2 Ω | − | ≤ | − | Z 1 for any η (0, η ]. Since n 2, 3 , the choice η = min η , t t 7 1 leads to ∈ 0 ∈ { } { 0 | 2 − 1| }≤

2 1 −3 1 ρε(t2) ρε(t1) dx C t2 t1 14 + η t2 t1 2 Ω | − | ≤ | − | | − | Z 1 −3 3 C t t 14 1+ η t t 7 ≤ | 2 − 1| | 2 − 1| 1 −1 1 3 C t t 14 1+ η t t 7 . ≤ | 2 − 1| | 2 − 1| 1 −1 1 −1 Finally, the choice of η gives η−1 t t 7 = max η t t 7 , 1 max η , 1 , which | 2 − 1| { 0 | 2 − 1| }≤ { 0 } implies (3.3.18).

Step 3. To verify (3.3.19), we use the deﬁnition of rε and the growth estimate for Φ, see (3.3.4) and (3.3.5), to conclude that

rε(t ) rε(t ) dx | 2 − 1 | ZΩ = Φ(ρε(t )) Φ(ρε(t )) dx | 2 − 1 | ZΩ C ρε(t ) ρε(t ) ρε(t ) a + ρε(t ) a + b dx ≤ 2 | 2 − 1 | | 2 − | | 1 − | ZΩ 1 C ρε(t ) ρε(t ) 2 ρε(t ) a 2 + ρε(t ) a 2 + b Ω 2 . ≤ 2k 2 − 1 kL (Ω) k 2 − kL (Ω) k 1 − kL (Ω) | | Using (3.3.17) and (3.3.18), we ﬁnally obtain (3.3.19). This concludes the proof.

55 Chapter 3 The Diﬀuse-Interface Model

3.4 Compactness of Weak Solutions

In this section we will use the a priori estimates from the previous section to prove suitable

compactness results for families of weak solutions (ρε, vε)ε∈(0,1) to (3.2.1)–(3.2.6). As an important consequence of the a priori estimates proven in Theorem 3.3.2, we can extract

converging subsequences from the sequence (vε)ε∈(0,1).

Theorem 3.4.1 (Compactness of velocity). Suppose that Assumptions 3.3.1 hold true.

N N Let (vεj )j∈ be a subsequence of (vε)ε∈(0,1). Then there exist a subsequence (vεjm )m∈ and a function v L2(0, T ; H1(Ω)n) such that, as m , there holds 0 ∈ 0 →∞

2 1 n 2 6 n .( (vε ⇀ v weakly in L (0, T ; H (Ω) ) ֒ L (0, T ; L (Ω jm 0 →

2 1 n Proof. The sequence (vε)ε∈(0,1) is uniformly bounded in L (0, T ; H (Ω) ) due to The- orem 3.3.2. This yields the weak convergence property in L2(0, T ; H1(Ω)n). Moreover, in view of Assumptions 3.2.1, we consider dimension n = 2, 3. Hence the embedding

(L2(0, T ; H1(Ω)n) ֒ L2(0, T ; L6(Ω)n →

2 1 n is continuous. As the space L (0, T ; H0 (Ω) ) is closed under weak convergence and as 2 1 n 2 1 n (vε) L (0, T ; H (Ω) ), by Deﬁnition 3.2.5, we conclude that v L (0, T ; H (Ω) ). ε∈(0,1) ⊂ 0 0 ∈ 0

To extract converging subsequences from (ρε)ε∈(0,1), it is more convenient to study the

transformed sequence (rε)ε∈(0,1), see (3.3.4), instead of directly considering (ρε)ε∈(0,1). This

is because, there are uniform bounds on ( rε) available, see Lemma 3.3.6, which is ∇ ε∈(0,1) not the case for ( ρε) . In addition, we will make use of the H¨older continuity of ∇ ε∈(0,1) (rε)ε∈(0,1) with respect to the time variable, which follows from Theorem 3.3.11. To be

precise, we will show that the sequence (rε)ε∈(0,1) is uniformly bounded in

∞ 0, 1 1 L (0, T ; BV (Ω)) C 28 ([0, T ]; L (Ω)). ∩ The following auxiliary lemma will then allow us to extract converging subsequences. Its proof is based on an abstract compactness result due to Simon [90], which we adapt to our situation.

Lemma 3.4.2 (Embedding into H¨older spaces). Let 0 <α<β and let X,Y be Banach . spaces such that X ֒ ֒ Y →→ 0,β 0 1. Let (fk)k N C ([0, T ]; Y ) be a bounded sequence and let f C ([0, T ]; Y ). If fk f ∈ ⊂ ∈ → 0 0,β 0,α in C ([0, T ]; Y ) for k , then f C ([0, T ]; Y ), and fk f in C ([0, T ]; Y ) as → ∞ ∈ → k . →∞ .The embedding L∞(0, T ; X) C0,β([0, T ]; Y ) ֒ ֒ C0,α([0, T ]; Y ) is compact .2 ∩ →→

56 3.4 Compactness of Weak Solutions

0 Proof. 1. Let 0 t

f(t2) f(t1) Y = lim fk(t2) fk(t1) Y . k − k k→∞ k − k

0,β By assumption, the sequence (fk)k∈N is uniformly bounded in C ([0, T ]; Y ). Therefore, we obtain

f(t2) f(t1) Y fk(t2) fk(t1) Y k − β k = lim k − β k lim sup fk C0,β ([0,T ];Y ) C. t t k→∞ t t ≤ k→∞ k k ≤ | 2 − 1| | 2 − 1| Taking the supremum over all 0 t < t T implies that f C0,β([0, T ]; Y ). To ≤ 1 2 ≤ ∈ prove convergence in C0,α([0, T ]; Y ), it only remains to verify that

(fk f)(t2) (fk f)(t1) Y lim sup k − − α− k = 0. (3.4.1) k→∞ t 0, 1 2 ∈ | 2 − 1| ≤ which we will later choose independently of k N. It then follows that ∈

(fk f)(t2) (fk f)(t1) Y (fk f)(t2) (fk f)(t1) Y β−α k − − α− k = k − − β− k t2 t1 t2 t1 t t | − | | − | | 2 − 1| β−α fk f 0,β δ . ≤ k − kC ([0,T ];Y )

Secondly, for all t ,t [0, T ] such that t t δ, we have 1 2 ∈ | 2 − 1|≥

(fk f)(t2) (fk f)(t1) Y −α k − − − k (fk f)(t ) (fk f)(t ) Y δ t t α ≤ k − 2 − − 1 k | 2 − 1| −α 2 fk f 0 δ . ≤ k − kC ([0,T ];Y )

Altogether, we obtain

(fk f)(t2) (fk f)(t1) Y β−α −α sup k − − α− k Cδ + 2δ fk f C0([0,T ];Y ). t 0 small enough, and then choosing k N large enough, we ∈ 0,α infer that fk f in C ([0, T ]; Y ) as k . → →∞ ∞ 0,β 2. Let (fk)k N L (0, T ; X) C ([0, T ]; Y ) be a bounded sequence. As the embed- ∈ ⊂ ∩ ding L∞(0, T ; X) C0,β([0, T ]; Y ) ֒ ֒ C0([0, T ]; Y ) is compact [90, Theorem 3], we ∩ →→ may extract a subsequence from (fk)k N, still denoted by (fk)k N, such that fk f ∈ ∈ → 0 0 in C ([0, T ]; Y ) as k , for some f C ([0, T ]; Y ). As (fk)k N is bounded in → ∞ ∈ ∈ C0,β([0, T ]; Y ), the ﬁrst statement of this lemma yields that f C0,β([0, T ]; Y ) and ∈ 0,α fk f in C ([0, T ]; Y ) for k . → →∞ This concludes the proof.

The a priori estimates for the sequence (rε)ε∈(0,1), proven in Lemma 3.3.6 and The- orem 3.3.11, lead to the following compactness properties.

57 Chapter 3 The Diﬀuse-Interface Model

Corollary 3.4.3 (Compactness of rε). Suppose that Assumptions 3.3.1 hold true and let

(rε)ε∈(0,1) be as in (3.3.4). Then, for any subsequence (rεj )j∈N of (rε)ε∈(0,1), there exist a ∞ 0, 1 1 subsequence (rε )m N and a limit function r L (0, T ; BV (Ω)) C 28 ([0, T ]; L (Ω)) jm ∈ 0 ∈ ∩ with the following convergence properties for m . →∞

0, 1 1 1. rε r in C 29 ([0, T ]; L (Ω)). jm → 0

2. r (t) (Ω) lim infm rε (t) (Ω) for every t [0, T ]. |∇ 0 | ≤ →∞ ∇ jm ∈

Proof. During the proof we will not relabel subsequences. In view of Lemma 3.3.6 and Theorem 3.3.11, there holds

∞ 1 rε L (0,T ;BV (Ω)) + rε 0, C. (3.4.2) k k k kC 28 ([0,T ];L1(Ω)) ≤

In view of Theorem 2.3.5, the embedding BV (Ω) ֒ ֒ L1(Ω) is compact. Hence, by →→ ∞ 0, 1 1 0, 1 1 Lemma 3.4.2, so is L (0, T ; BV (Ω)) C 28 ([0, T ]; L (Ω)) ֒ ֒ C 29 ([0, T ]; L (Ω)). This ∩ →→ 0, 1 1 allows us to extract a subsequence such that rε r in C 29 ([0, T ]; L (Ω)) as ε 0, where → 0 → 0, 1 1 the limit function r0 belongs to C 28 ([0, T ]; L (Ω)), again by Lemma 3.4.2. In particular, 1 for any t [0, T ], there holds rε(t) r (t) in L (Ω) as ε 0. By Theorem 2.3.6, the ∈ → 0 → total variation is lower semi-continuous with respect to strong L1-convergence, hence we infer that

r0(t) (Ω) lim inf rε(t) (Ω). |∇ | ≤ ε→0 |∇ | This, in turn, implies that r L∞(0, T ; BV (Ω)) because of (3.4.2). 0 ∈

We collect the compactness properties of (ρε)ε∈(0,1) in the following theorem.

Theorem 3.4.4 (Compactness of density). Suppose that Assumptions 3.3.1 hold true.

N N For any subsequence (ρεj )j∈ of (ρε)ε∈(0,1), there exist a subsequence (ρεjm )m∈ and a 0, 1 2 function ρ C 28 ([0, T ]; L (Ω)) such that, for every t [0, T ], the following convergence 0 ∈ ∈ properties are valid as m . →∞

0, 1 2 1. ρε ρ in C 29 ([0, T ]; L (Ω)). jm → 0

p∗ 2. ρεjm (t) ⇀ ρ0(t) weakly in L (Ω).

q 3. ρε (t) ρ (t) in L (Ω) for any q [1,p ). jm → 0 ∈ ∗ Proof. During the proof we will not relabel subsequences.

1. Let (rε) be as in (3.3.4). Due to Corollary 3.4.3, as ε 0, for some limit function ε∈(0,1) → 0, 1 1 0, 1 1 r C 28 ([0, T ]; L (Ω)), there holds rε r in C 29 ([0, T ]; L (Ω)). Note that, in 0 ∈ → 0 particular, for all t [0, T ], after possibly passing to a suitable subsequence depending ∈ on t, we have rε(t) r (t) a.e. in Ω as ε 0. → 0 →

58 3.4 Compactness of Weak Solutions

Since ρε(t) 0 a.e. in Ω and by Lemma 3.3.3, there holds rε(t) Φ(0) a.e. in Ω for ≥ ≥ all ε (0, 1). Hence r (t) Φ(0) a.e. in Ω. Deﬁning ρ = Φ−1 r , we obtain that ∈ 0 ≥ 0 ◦ 0 ρ (t) 0 a.e. in Ω. 0 ≥ By Lemma 3.3.3 and Proposition 3.3.4, the transformation Φ, see (3.3.3) and (3.3.7), is invertible and satisﬁes, for all s ,s [Φ(0), ), the growth estimate 1 2 ∈ ∞ 2 Φ−1(s ) Φ−1(s ) C s s , 2 − 1 ≤ | 2 − 1|

−1 for some constant C > 0. As ρε =Φ rε, we therefore infer that ◦

2 ρε(t) ρ (t) dx C rε(t) r (t) dx | − 0 | ≤ | − 0 | ZΩ ZΩ 0 2 for any t [0, T ]. Hence we obtain ρε ρ in C ([0, T ]; L (Ω)) as ε 0. ∈ → 0 → 0, 1 2 Since, by Theorem 3.3.11, (ρε)ε>0 is uniformly bounded in C 28 ([0, T ]; L (Ω)), we may 0, 1 2 apply Lemma 3.4.2, which implies that ρ C 28 ([0, T ]; L (Ω)) and, for ε 0, there 0 ∈ → holds 0, 1 2 ρε ρ in C 29 ([0, T ]; L (Ω)). (3.4.3) → 0

2. Let t [0, T ] be ﬁxed. By (3.4.3), the continuity of W and Lemma 3.1.4, (ρε(t)) ∈ ε∈(0,1) is uniformly bounded (in ε) in Lp∗ (Ω). Consequently, in order to show that

p∗ ρε(t) ⇀ ρ0(t) weakly in L (Ω) (3.4.4)

as ε 0, it suﬃces to prove that for any subsequence (ρε (t))m N such that → m ∈

p∗ ρεm (t) ⇀ ρ weakly in L (Ω)

p∗ as m , for some ρ L (Ω), we have ρ = ρ (t). For this reason, let (ρε (t))m N → ∞ ∈ 0 m ∈ be such a subsequence. On the one hand, as p∗ > 2, there holds ρεm (t) ⇀ ρ weakly in 2 L (Ω) as m . On the other hand, in view of (3.4.3), we have ρε (t) ρ (t) in → ∞ m → 0 L2(Ω) for m . Hence ρ = ρ (t), and (3.4.4) follows. →∞ 0 3. Fix t [0, T ]. We need only consider the case q (2,p ). Then, by the intermediate- ∈ ∈ ∗ value theorem, there exists some α (0, 1) such that 1 = α + 1−α . By interpolation, ∈ q 2 p∗ we obtain

α 1−α ρε(t) ρ (t) q ρε(t) ρ (t) 2 ρε(t) ρ (t) p∗ k − 0 kL (Ω) ≤ k − 0 kL (Ω)k − 0 kL (Ω) α 1−α ρ ρ 1 ρ (t) ρ (t) . ε 0 0, ε 0 Lp∗ (Ω) ≤ k − kC 29 ([0,T ];L2(Ω))k − k

p∗ The weak L -convergence (3.4.4) implies boundedness of ρε(t) ρ (t) p∗ . Finally, k − 0 kL (Ω) q (3.4.3) implies that ρε(t) ρ (t) in L (Ω) as ε 0. → 0 → This completes the proof.

59 Chapter 3 The Diﬀuse-Interface Model

3.5 Limiting Interfaces

In the foregoing section we proved compactness properties of weak solutions (ρε, vε)ε∈(0,1) to the diﬀuse-interface model (3.2.1)–(3.2.6). In particular, in Theorem 3.4.4, we showed that ρε ρ as ε 0 in suitable function spaces, at least along appropriate subsequences. → 0 → In this section we shall show that, for any t [0, T ], the limit function ρ (t) only takes ∈ 0 values in β , β a.e. and is of bounded variation. Subsequently, we will use this regularity { 1 2} property to deﬁne a corresponding (sharp) interface Γ(t) induced by ρ0(t).

3.5.1 Characteristic Functions

The ﬁrst step towards a limiting interface is the study of characteristic (or indicator) n functions χE of some subset E of R , i.e.,

1 for x E, χE(x)=  ∈ 0 for x Rn E.  ∈ \ For a set E Ω, there hold the equivalences ⊂ ∞ E is measurable χE is measurable χE L (Ω) ⇐⇒ ⇐⇒ ∈ and

∞ E is of ﬁnite perimeter in Ω χE BV (Ω) χE BV (Ω) L (Ω). ⇐⇒ ∈ ⇐⇒ ∈ ∩

The characteristic function χE represents the set E, and vice versa. Considering characteristic functions as elements of Lebesgue spaces suggests the following questions.

1. Is every measurable function χ with χ 0, 1 almost everywhere a characteristic ∈ { } function? That is, does there exist a measurable set E representing χ in the sense that

that χ and χE coincide almost everywhere?

2. Is χE invariant in the equivalence class

[E] n = F Ω : F measurable, (E F ) (E F ) = 0 L ⊂ | ∪ \ ∩ | of E with respect to the Lebesgue measure, i.e., for every F [E] n , do the correspond- ∈ L ing characteristic functions χE and χF coincide almost everywhere? If so, it is natural

to ask for “good” representatives of the class [E]Ln .

The following lemma answers the first question in the affirmative, in a constructive manner, cf. also Definition 2.3.11.

Lemma 3.5.1. Suppose that Assumptions 3.2.1 hold true. Let χ L∞(Ω) be such that ∈ χ(x) 0, 1 for a.e. x Ω. Then the sets ∈ { } ∈ E− = x Ω : lim 1 χ(y)dy = 1 ∈ δ→0 |Bδ (x)| ( ZBδ(x) )

60 3.5 Limiting Interfaces and E+ = x Ω : lim 1 χ(y)dy = 0 ∈ δ→0 |Bδ (x)| ( ZBδ (x) ) are measurable. Furthermore, there holds

χ = χ − = 1 χ + a.e. in Ω. (3.5.1) E − E If, additionally, χ BV (Ω, 0, 1 ), then the sets E− and E+ have ﬁnite perimeter. ∈ { } Proof. It suﬃces to prove (3.5.1). Since χ L1(Ω), a.e. x Ω is a Lebesgue point, see for ∈ ∈ example [16, Corollary 2.23]; that is,

χ(x) = lim 1 χ(y)dy. δ→0 |Bδ(x)| ZBδ(x) Now the identity (3.5.1) follows, since χ only takes values in 0, 1 a.e. in Ω. { } To answer the second question concerning the representative set of a characteristic function, we shall in a certain sense invert the construction from Lemma 3.5.1. The measure- theoretic interior, see Deﬁnition 2.3.11, turns out to be appropriate for our purposes.

Theorem 3.5.2. Let E Rn be a measurable set. Then the measure-theoretic interior ⊂ 1 0 E , the measure-theoretic exterior E and the essential boundary ∂∗E are invariants of the n equivalence class [E] n ; that is, for any measurable set F R such that F [E] n , there L ⊂ ∈ L 1 1 0 0 holds F = E , F = E and ∂∗F = ∂∗E. Moreover, there holds χE1 = χE, χE0 = χRn\E Rn and χ∂∗E = 0 a.e. in .

Proof. See [32, Chapter 5, Theorem 3.3].

3.5.2 Representation of Limiting Density

In the sequel we are only concerned with (characteristic) functions of bounded variation. We give a slightly generalised version of the above Lemma 3.5.1 adapted to our situation.

Lemma 3.5.3. Suppose that Assumptions 3.2.1 hold true and let ρ L1(Ω). Then, for ∈ all b , b R, the following statements are equivalent. 1 2 ∈ 1. ρ BV (Ω, b , b ). ∈ { 1 2} 2. There exists a function χ BV (Ω, 0, 1 ) such that ρ = (b b )χ + b a.e. in Ω. ∈ { } 1 − 2 2

3. There exists a set E Ω of ﬁnite perimeter in Ω such that ρ = (b b )χE + b a.e. ⊂ 1 − 2 2 in Ω.

ρ−b2 Proof. If ρ BV (Ω, b1, b2 ), then the function χ = belongs to BV (Ω, 0, 1 ) and ∈ { } b1−b2 { } satisﬁes ρ = (b b )χ + b a.e. in Ω. The opposite direction is obvious. The equivalence 1 − 2 2 of the second and the third statement follows from Lemma 3.5.1.

61 Chapter 3 The Diﬀuse-Interface Model

Now the above lemma gives rise to the following deﬁnition.

Deﬁnition 3.5.4 (Measure-theoretic representative set). Let b , b R and suppose that 1 2 ∈ ρ BV (Ω, b , b ). ∈ { 1 2} 1. A function χ BV (Ω, 0, 1 ) is called an induced characteristic function of ρ if ρ = ∈ { } (b b )χ + b a.e. in Ω. 1 − 2 2 2. A set E Ω of ﬁnite perimeter in Ω is called a representative set of ρ if ρ = (b ⊂ 1 − b2)χE + b2 a.e. in Ω.

3. If E Ω is a representative set of ρ, then the measure-theoretic interior E− of E is ⊂ called the measure-theoretic representative set of ρ.

Remark 3.5.5. The notion of the measure-theoretic representative set is well-deﬁned because of Theorem 3.5.2.

Lemma 3.5.3 ﬁnally leads to the following representation of ρ0. Instead of directly con-

sidering ρ0, it is, once more, convenient to work with the limiting function r0 from Corol- lary 3.4.3.

Corollary 3.5.6 (Representation of r0 and ρ0). Let Assumptions 3.3.1 hold true and set

β2 σ = Φ(β ) Φ(β )= min 1 W (z), z a 2 + b2 dz, (3.5.2) st 2 − 1 2 | − | Zβ1 q where Φ is given by (3.3.3). Then the limit functions r0 and ρ0 from Corollary 3.4.3 and Theorem 3.4.4 have, for all t [0, T ], the following properties. ∈ 1. r (t) BV (Ω, Φ(β ), Φ(β ) ) and ρ (t) BV (Ω, β , β ). 0 ∈ { 1 2 } 0 ∈ { 1 2}

2. The measure-theoretic representative sets of r0(t) and ρ0(t) coincide, and for the common measure-theoretic representative set Ω−(t), there holds

r (t)= σ χ − + Φ(β ) and ρ (t) = (β β )χ − + β a.e. in Ω. 0 − st Ω (t) 2 0 1 − 2 Ω (t) 2

In particular, r L∞(0, T ; BV (Ω, Φ(β ), Φ(β ) )) and ρ L∞(0, T ; BV (Ω, β , β )). 0 ∈ { 1 2 } 0 ∈ { 1 2}

2 Proof. Let t [0, T ]. By Theorem 3.4.4, ρε(t) ρ (t) in L (Ω) as ε 0 and, additionally, ∈ → 0 → ρε(t) ρ (t) a.e. in Ω after passing to a subsequence, which we do not relabel. Notice → 0 that the choice of the subsequence now depends on t.

The continuity of W yields W (ρε(t)) W (ρ (t)) a.e. in Ω for ε 0. As W 0, Fatou’s → 0 → ≥ lemma applies, and we obtain

tot 0 W (ρ0(t)) dx lim inf W (ρε(t)) dx lim inf εEε (t). ≤ ≤ ε→0 ≤ ε→0 ZΩ ZΩ

62 3.6 Remarks

By the energy inequality (3.2.16) and by Assumptions 3.3.1, it follows that

W (ρ0(t)) dx = 0. ZΩ We therefore have W (ρ (t)) = 0 a.e. in Ω, or equivalently, ρ (t) β , β a.e. in Ω. By 0 0 ∈ { 1 2} Corollary 3.4.3, we conclude that r (t)=Φ(ρ (t)) belongs to BV (Ω, Φ(β ), Φ(β ) ). 0 0 { 1 2 } For the proof of the remaining claims, in view of Lemma 3.5.3, it only remains to verify that r0(t) and ρ0(t) have the asserted representations: Lemma 3.5.3 now implies the existence of a set Ω−(t) of ﬁnite perimeter in Ω such that

r (t) = (Φ(β ) Φ(β ))χ − + Φ(β )= σ χ − + Φ(β ) a.e. in Ω. 0 1 − 2 Ω (t) 2 − st Ω (t) 2 −1 Hence, from ρ = Φ r , we see that ρ (t) = (β β )χ − + β a.e. in Ω. This 0 ◦ 0 0 1 − 2 Ω (t) 2 concludes the proof.

Remark 3.5.7 (Canonical double-well potential). Note that in the case of the canonical double-well potential W (z) = (z β )2(z β )2 = (z a)2 b2 2, where a = β1+β2 − 1 − 2 − − 2 β2−β1 and b = 2 , which we already introduced in Example 3.1.3, the transfor mation Φ, see (3.3.3), simpliﬁes to

s s Φ(s)= min 1 W (z), z a 2 + b2 dz = 1 W (z)dz 2 | − | 2 Za Za q q for s [0, ). In turn, for the constant σ , see (3.5.2), we have ∈ ∞ st β2 1 σst = Φ(β2) Φ(β1)= 2 W (z)dz. − β Z 1 q 3.5.3 Induced Partition of Domain and Limiting Sharp Interface

By the preceding results, for every t [0, T ], ρ (t) BV (Ω, β , β ) induces a disjoint ∈ 0 ∈ { 1 2} partition of the domain Ω. More precisely, the measure-theoretic representative set Ω−(t) of ρ0(t), see Deﬁnition 3.5.4, induces the disjoint partition

Ω=Ω−(t) Γ(t) Ω+(t), ∪ ∪ where the (sharp) interface Γ(t) and Ω+(t) are, respectively, deﬁned by

Γ(t)= ∂ (Ω−(t)) Ω and Ω+(t)=Ω (Ω−(t) Γ(t)). (3.5.3) ∗ ∩ \ ∪ Hence we have established the existence of a limiting sharp interface. This is the ﬁrst step towards the sharp-interface limit.

3.6 Remarks

To conclude the present chapter, we review the previous results regarding the sharp- interface limit for weak solutions (ρε, vε)ε∈(0,1) of the Navier–Stokes–Korteweg system.

63 Chapter 3 The Diﬀuse-Interface Model

The compactness results from Theorems 3.4.1 and 3.4.4 allow us to extract subsequences

from (ρε, vε)ε∈(0,1) converging to limit functions (ρ0, v0). Furthermore, as explained in

Section 3.5.3, any limiting density ρ0 induces a sharp interface. At this point the natural question is: to what extent do these results allow us to pass to the limit ε 0 in the weak formulation of the system (3.2.1)–(3.2.6) given in Definition 3.2.5? → The answer to this question will strongly influence the subsequent analysis. It turns out that the results of the present chapter are not sufficient to control the limiting

behaviour of (ρε, vε) for ε 0. In particular, they are not suﬃcient to control the ε∈(0,1) → limiting behaviour of the terms

1 T T p(ρε) div(ψ)dx dt and ε ρε ρε : ψ dx dt (3.6.1) ε ∇ ⊗∇ ∇ Z0 ZΩ Z0 ZΩ in the weak formulation of the linear-momentum equation (3.2.15). These diﬃculties

stem from a lack of control on ( ρε) . As the limiting density ρ (t) is a function of ∇ ε∈(0,1) 0 bounded variation “jumping” between the values β1 and β2, see Corollary 3.5.6, we expect

(heuristically) ρε to have “steep” gradients for suﬃciently small ε and that ρ0(t) is not

weakly diﬀerentiable. Therefore, we cannot hope for convergence of ( ρε) , say in ∇ ε∈(0,1) r 2 some L -space, even though we have a uniform L -bound on (√ε ρε) , which follows ∇ ε∈(0,1) from the energy estimate (3.2.16). A major diﬃculty in performing the sharp-interface limit is to control the convergence of the two terms in (3.6.1). Our strategy for treating these expressions will be twofold.

1. As we will see later in the proof of Theorem 5.4.2, the results of Theorems 3.4.1 and 3.4.4 allow us to pass to the limit ε 0 in the weak formulation of the conservation-of-mass → equation (3.2.14). Assume for the moment that the limiting mass conservation

∂tρ + div(ρ v )=0inΩ (0, T ) 0 0 0 ×

− holds true, and that the measure-theoretic representative set Ω (t) of ρ0(t) as well as + − the set Ω (t), deﬁned in (3.5.3), are open subsets of Ω. As ρ0(t) is constant in Ω (t) and Ω+(t), respectively, we obtain the incompressibility conditions

− + div(v0(t))=0in Ω (t) and div(v0(t))=0 in Ω (t).

Similar as in the theory of the incompressible Navier–Stokes equations, this gives rise to a sharp-interface model incorporating a weak formulation which uses divergence- free test functions. The sharp-interface model is the content of Chapter 4, where, in particular, we shall derive such a weak formulation. This proceeding resolves the diﬃculty of controlling 1 T p(ρ ) div(ψ)dx dt, ε ε Z0 ZΩ as this expression vanishes for divergence-free test functions ψ.

64 3.6 Remarks

2. In Section 5.1.2 we will assume convergence of an associated energy functional to a suitable surface measure. This condition will turn out to be suﬃcient to control the term T ε ρε ρε : ψ dx dt. ∇ ⊗∇ ∇ Z0 ZΩ This convergence property and the meaning of the limiting surface measure will be discussed more thoroughly in Section 5.7.

Chapter 4

The Sharp-Interface Model

In this chapter we are concerned with the two-phase incompressible Navier–Stokes equations with surface tension (1.4.9)–(1.4.14) that will turn out to be later in Chapter 5 the sharp-interface limit of the phase-field-like scaling of the Navier–Stokes–Korteweg equations presented in Chapter 3. For this free-boundary problem, we seek to derive an appropriate weak formulation. For this purpose, we will generalise ideas of Abels [1], who investigated the case of two fluids with equal densities β1 = β2 = 1, to the case of two fluids of different densities β β . Subsequently, we will prove a consistency result to 1 ≤ 2 justify our definition. This chapter is organised as follows: in Section 4.1 we will introduce the free-boundary problem. During this chapter we will need some geometric results which we will briefly summarise in Section 4.2. As already announced in Section 3.6, in Section 4.3 we will derive a weak notion of solutions which uses divergence-free test functions. This will lead to a weak formulation that does not incorporate the pressure function. In the remainder of the present chapter we shall justify our approach and reconstruct a pressure function from the weak formulation: in Section 4.4 we shall provide the functional-analytic background and introduce Sobolev spaces on time-dependent domains. In Section 4.5, under reasonable regularity assumptions, we will reconstruct the pressure function from the weak formulation.

4.1 Two-Phase Incompressible Navier–Stokes Equations with Surface Tension

The isothermal flow of a Newtonian fluid in a domain Ω Rn and in a finite time interval ⊂ [0, T ] is described in Eulerian coordinates by a velocity field v :Ω [0, T ] Rn and a × → scalar pressure function p:Ω [0, T ] R. For each time t [0, T ], a hypersurface Γ(t) × → ∈ separates Ω into two disjoint open subsets Ω−(t) and Ω+(t) of Ω, i.e., we have

Ω=Ω−(t) Γ(t) Ω+(t) and Γ(t)= ∂Ω−(t) Ω. ∪ ∪ ∩ The regions Ω−(t) and Ω+(t) are referred to as bulk phases and correspond to the vapour and the liquid phase of the ﬂuid. Physically they are characterised by diﬀerent (con-

67 Chapter 4 The Sharp-Interface Model

stant) densities β1 < β2 and corresponding viscosities µ(β1) and µ(β2). For convenience, throughout this section we will require that the interface is compactly contained in the fluid domain, that is, Γ(t) Ω. In particular, the interface does not intersect with the ⊂⊂ domain’s boundary, i.e., Γ(t) ∂Ω= . This, in turn, means that ∩ ∅ Ω−(t) Ω and Γ(t)= ∂Ω−(t)= ∂Ω−(t) ∂Ω+(t). (4.1.1) ⊂⊂ ∩ Assuming the interface Γ to be sufficiently regular, and the velocity v and the pressure p to be sufficiently smooth functions on Ω Γ(t)=Ω−(t) Ω+(t) (the precise assumptions will \ ∪ be given in Assumptions 4.3.1 below), the flow is described by the following free-boundary problem

− β ∂tv + β (v )v µ(β )∆v + p =0 inΩ (t), (4.1.2) 1 1 ·∇ − 1 ∇ + β ∂tv + β (v )v µ(β )∆v + p =0 inΩ (t), (4.1.3) 2 2 ·∇ − 2 ∇ div(v)=0 inΩ Γ(t), (4.1.4) \ [v]=0 onΓ(t), (4.1.5) V = v ν− on Γ(t), (4.1.6) · [T ] ν− = 2σ κν− on Γ(t), (4.1.7) − st v( ,t)=0 on ∂Ω, (4.1.8) · v( , 0) = v(i) in Ω (4.1.9) · for every t [0, T ]. Given are the initial phases Ω−(0) = Ω−,(i) Ω,Ω+(0) = Ω Ω−,(i) ∈ ⊂⊂ \ and the initial position Γ(i) = ∂(Ω−,(i)) of the interface as well as the initial velocity v(i) :Ω Rn. The unknowns are the velocity v( ,t):Ω Γ(t) Rn, the pressure p( ,t):Ω → · \ → · \ Γ(t) R and the interface (free-boundary) Γ(t). Here and in the sequel, we will use the → notation [ ] for the jump across the interface Γ(t) in the direction of the exterior unit- · normal ﬁeld ν−( ,t) of ∂Ω−(t). For a given quantity f and x Γ(t), this is explicitly · ∈ [f](x,t) = lim f(x + ξν−(x,t),t) f(x ξν−(x,t),t) . ξց0 − − By V and κ, we denote the normal velocity and the mean curvature of Γ, both taken with respect to ν−. Exact deﬁnitions of these and other geometric quantities are summarised in

Section 4.2. Moreover, σst > 0 denotes the surface-tension constant, and the stress tensor T = T (v,p) in (4.1.7) is deﬁned by

− 2µ(β1)Dv(t) p(t)I in Ω (t), T (v(t),p(t)) =  − 2µ(β )Dv(t) p(t)I in Ω+(t).  2 − The partial diﬀerential equations (4.1.2)–(4.1.4) are the incompressible Navier–Stokes equations. Equations (4.1.2) and (4.1.3) model the conservation of linear momentum and the incompressibility condition (4.1.4) corresponds to conservation of mass in each phase.

68 4.1 Two-Phase Incompressible Navier–Stokes Equations with Surface Tension

The partial differential equations in the bulk phases are coupled by the interface conditions (4.1.5)–(4.1.7): the velocity field is continuous across the interface Γ(t) by (4.1.5). Due to (4.1.6), the interface Γ(t) is purely transported with the flow of the mass particles. The interface condition (4.1.7) is (a dynamic version of) the Young–Laplace law relating the pressure jump [p] to the mean curvature κ. The velocity boundary condition (4.1.8) is the no-slip condition at the boundary ∂Ω of the fluid domain Ω. By (4.1.9), we prescribe initial values v(i) :Ω Rn for the velocity. → We conclude the introduction of the sharp-interface model with the following remarks.

Remark 4.1.1 (Incompressible phases). Due to the incompressibility condition (4.1.4), there holds div(v) = 0 in Ω Γ(t). In the sequel we will often use the relations \

div(v v) = div(v)v + (v )v = (v )v in Ω Γ(t) (4.1.10) ⊗ ·∇ ·∇ \ and

2 div(Dv)=∆v + div(v)=∆v in Ω Γ(t). (4.1.11) ∇ \

Remark 4.1.2. The choice that the phase Ω−(t) is compactly contained in Ω, see (4.1.1), is arbitrary and only for notational convenience. All the results of the present chapter are also valid in the case Ω+(t) Ω. ⊂⊂

Remark 4.1.3 (Existence of solutions). The question of (unique) solvability of the free- boundary problem (4.1.2)–(4.1.9) and related systems has been studied by many authors. In the framework of Hölder spaces, Denisova and Solonnikov first studied the corresponding two-phase Stokes problem [33]. Later they proved well-posedness of (4.1.2)–(4.1.9) for appropriate initial data [34]. Existence results in the context of maximal Lr-regularity are due to Prüss and Simonett [82] and Köhne, Prüss and Wilke [59].

Remark 4.1.4 (Exclusion of phase transitions). In view of the interface condition (4.1.6), the interface velocity and that of the ﬂuid coincide in the normal direction. This means that the interface is purely transported by the ﬂuid and no phase transition occurs. For the case of phase transitions we refer to [80, 81].

Remark 4.1.5 (Surface-tension constant). With regard to the sharp-interface limit for the phase-ﬁeld-like scaling of the Navier–Stokes–Korteweg equations, it will turn out that the surface-tension constant σst is given by

β2 σ = min 1 W (z), z a 2 + b2 dz, st 2 | − | Zβ1 q cf. (1.5.6) and (3.5.2).

69 Chapter 4 The Sharp-Interface Model

4.2 Hypersurfaces

The sharp-interface model (4.1.2)–(4.1.9) incorporates geometric quantities such as the mean curvature and the normal velocity of the interface Γ(t). Here, we give precise definitions and briefly recall some facts from differential geometry. For a fuller treatment we refer for instance to [31, Section 2], [48, Section 16.1] and [58]. For our purposes, it is convenient to focus on hypersurfaces embedded in the d-dimensional Euclidean space Rd. Throughout this section it is assumed that d 2. ≥ Definition 4.2.1 (Hypersurfaces). Let Γ Rd. ⊂ 1. For k N, Γ is called a Ck-hypersurface if, for each x Γ, there exist an open ∈ 0 ∈ neighbourhood U Rd of x and a function u Ck(U) with ⊂ 0 ∈ U Γ= x U : u(x) = 0 and u(x) = 0 for all x U Γ. ∩ { ∈ } ∇ 6 ∈ ∩

The function u is called a (local) level-set function of Γ in x0.

2. If Γ is a C2-hypersurface, it is (brieﬂy) called a hypersurface.

3. If Γ is a hypersurface, then the space C1(Γ) consists of all f : Γ R such that there → exist a neighbourhood U Rd of Γ and a function g C1(U) with f = g . ⊂ ∈ |U d 4. The tangent space TxΓ of a hypersurface Γ R , in a point x Γ, is the (d 1)- ⊂ ∈ − dimensional subspace of Rd that is orthogonal to u(x), i.e., ∇ d TxΓ= τ R : τ u(x) = 0 . { ∈ ·∇ }

5. A hypersurface Γ is called oriented if there exists a function ν C1(Γ)d such that, for ∈ all x Γ, there holds ν(x) = 1 and ν(x) TxΓ, i.e., ν(x) τ = 0 for any τ TxΓ. ∈ | | ⊥ · ∈ The function ν is called unit-normal ﬁeld (or, brieﬂy, normal).

Remark 4.2.2. The deﬁnition of the tangent space does not depend on the choice of the level-set function u describing the hypersurface.

Definition 4.2.3 (Differential operators on hypersurfaces). Let Γ Rd be an oriented ⊂ hypersurface with unit-normal field ν.

1. For a function f C1(Γ), the tangential gradient f : Γ Rd is deﬁned as ∈ ∇Γ →

f = (δ f,...,δdf)= f ( f ν)ν . ∇Γ 1 ∇ − ∇ · Γ

2. For a function u C1(Γ)d, the tangential divergence div (u): Γ R is deﬁned as ∈ Γ → d divΓ(u)= δiui. Xi=1

70 4.2 Hypersurfaces

Remark 4.2.4. 1. The deﬁnitions of the tangential gradient f and the tangential di- ∇Γ vergence divΓ(u) depend only on the values of f and u, respectively, on Γ.

2. The tangential gradient f is the orthogonal projection of the gradient f onto the ∇Γ ∇ tangent space TxΓ. In particular, f is orthogonal to the normal ν; that is, for all ∇Γ x Γ, there holds ∈ f(x) ν(x) = 0. (4.2.1) ∇Γ · 3. There holds div (u) = div(u) u : ν ν by the definition of the tangential divergence. Γ −∇ ⊗ Proposition 4.2.5. For an oriented hypersurface Γ Rd with unit-normal field ν, define ⊂ d×d = ( ij)i,j ,...,d : Γ R by K K =1 →

ij(x)= δiνj(x) (4.2.2) K − for i, j = 1,...,d and x Γ. Then, for every x Γ, the matrix (x) is symmetric and ∈ ∈ K ν(x) is an eigenvector of (x) with corresponding eigenvalue 0. K Proof. See [31, Section 2.3] or [58, Theorem 2.10].

The foregoing proposition allows one to deﬁne the mean curvature of an oriented hypersurface.

Definition 4.2.6 (Mean curvature). For an oriented hypersurface Γ Rd with unit- ⊂ normal field ν, let x Γ and let be defined as in (4.2.2). ∈ K

1. The principal curvatures of Γ in x are the eigenvalues κ (x),...,κd (x) of (x) to 1 −1 K eigenvectors orthogonal to ν(x).

2. The mean curvature κ: Γ R is the trace of , i.e., → K d d−1 κ = ii = κi. K Xi=1 Xi=1 The mean curvature and the unit normal of a hypersurface are related in the following sense.

Proposition 4.2.7. For an oriented hypersurface Γ Rd with unit-normal ﬁeld ν and ⊂ mean curvature κ, there holds κ(x)= div (ν)(x) for x Γ. − Γ ∈ Proof. Let x Γ. Due to the deﬁnitions of the mean curvature and the tangential diver- ∈ gence, there holds

d d κ(x)= ii(x)= δiνi(x)= div (ν)(x). K − − Γ Xi=1 Xi=1

71 Chapter 4 The Sharp-Interface Model

The sign of the mean curvature depends on the choice of the unit-normal field. Moreover, the sign in the definition of , see (4.2.2) is not consistent in the literature and is used K −K instead of . The unit sphere provides an illustrative example helping to understand the K definitions at hand.

Example 4.2.8 (Unit sphere). The (d 1)-dimensional unit sphere Sd−1 Rd is oriented − ⊂ by either of the unit-normal ﬁelds ν (x) = x and ν (x) = x. In case of the mean + − − curvature κ introduced in Deﬁnition 4.2.6, in view of Proposition 4.2.7, we see that either κ(x) = div (ν (x)) = (d 1) if we take the mean curvature with respect to ν or − Γ + − − + κ(x)= div (ν (x)) = d 1 if we conversely take the mean curvature with respect to ν . − Γ − − − For hypersurfaces, there is the following important variant of the integration-by-parts formula.

Theorem 4.2.9 (Integration by parts). Let Γ Rd be a compact oriented hypersurface ⊂ with unit-normal ﬁeld ν and mean curvature κ. For f C1(Γ) and i = 1,...,d, there ∈ holds d−1 d−1 δif d (x)= fκνi d (x). H − H ZΓ ZΓ Proof. See [48, Lemma 16.1].

For the treatment of time-dependent interfaces Γ(t), we need the notion of evolving hypersurface and have to deﬁne its normal velocity.

Deﬁnition 4.2.10 (Evolving hypersurfaces). Let I R be an interval. For a family ⊂ (Γ(t))t∈I of oriented hypersurfaces, deﬁne

Γ= Γ(t) t . t∈I × { } [ 2,1 1. (Γ(t))t∈I is called a C -family of evolving oriented hypersurfaces, or brieﬂy family of evolving hypersurfaces, if Γ is a C1-hypersurface in Rd+1 and there exists a function ν C1(Γ)d such that Γ(t) is oriented by ν( ,t) for every t I. ∈ · ∈

0 2. The normal velocity V C (Γ) of (Γ(t))t I at a point (x ,t ) Γ is given by ∈ ∈ 0 0 ∈

V (x ,t )= η′(t ) ν(x ,t ), 0 0 0 · 0 0

where η C1(I )d, for some subinterval I I with t I , such that η(t ) = x and ∈ 0 0 ⊂ 0 ∈ 0 0 0 η(t) Γ(t) for all t I . ∈ ∈ 0 Remark 4.2.11. The deﬁnition of the normal velocity V does not depend on the choice of the function η. Moreover, for any t I, there holds V ( ,t) C1(Γ(t)); see [58, The- ∈ · ∈ orem 5.5].

72 4.3 The Notion of Weak Solutions

Finally, we provide some transport identities for integrals, which allow one to calculate time derivatives of integrals over time-dependent domains and hypersurfaces.

Theorem 4.2.12 (Transport theorem). For some interval I R, let (Γ(t))t I be a family ⊂ ∈ of evolving hypersurfaces in the sense of Deﬁnition 4.2.10. In addition, for every t I, ∈ assume that Γ(t)= ∂Ω(t) for some open, bounded set Ω(t) Rd. Denote by ν = ν(t) the ⊂ unit-normal ﬁeld of Γ(t) pointing outward to Ω(t), by κ = κ(t) the mean curvature of Γ(t) and by V the normal velocity of (Γ(t))t∈I , respectively, with respect to ν(t).

1. If U Rd+1 is an open set such that ⊂ Ω(t) t U, × { } ⊂ t[∈I then, for every f C1(U), there holds ∈

d d−1 f dx = ∂tf dx + fV d (x). dt H ZΩ(t) ZΩ(t) ZΓ(t)

2. If f C1(Γ), then there holds ∈ d f d d−1(x) dt H ZΓ(t) d−1 d−1 d−1 = ∂tf d (x) fκV d (x)+ ( f ν)V d (x). H − H ∇ · H ZΓ(t) ZΓ(t) ZΓ(t) In particular, d d−1(Γ(t)) = κV d d−1(x). dtH − H ZΓ(t) Proof. See [31, Appendix] or [58, Theorems 6.1 and 6.4].

4.3 The Notion of Weak Solutions

We are concerned with a weak formulation for the sharp-interface model (4.1.2)–(4.1.9). Here, our proceeding is analogous to Section 3.2, where we motivated a weak formulation of the diffuse-interface model by studying sufficiently smooth solutions. The free-boundary problem (4.1.2)–(4.1.9) incorporates two disjoint subregions Ω−(t) and + Ω (t) of the domain Ω, where the fluid is of constant density β1 and β2, respectively. This means that the associated density function is given by

ρ(t)= β1χΩ−(t) + β2χΩ+(t) in Ω. (4.3.1)

Note that ρ(t) = (β β )χ − + β in Ω Γ(t). Moreover, the nature of ρ(t) is encoded 1 − 2 Ω (t) 2 \ in the characteristic function

ρ(t) β2 χ(t)= χ − = − , (4.3.2) Ω (t) β β 1 − 2

73 Chapter 4 The Sharp-Interface Model

and vice versa, cf. Lemma 3.5.3. In many situations, it is convenient to use that (4.1.2) and (4.1.3) are equivalent to

ρ∂tv + ρ(v )v 2µ(ρ) div(Dv)+ p =0inΩ Γ(t). (4.3.3) ·∇ − ∇ \

To motivate a weak formulation, we consider suﬃciently smooth solution triplets (v,p, Γ) of the free-boundary problem (4.1.2)–(4.1.9). (Precise assumptions will be given below in Assumptions 4.3.1.) More precisely, for the pair (ρ, v), we derive the following statements:

1. a variational formulation for the identity (4.3.3) incorporating divergence-free test functions; see Lemma 4.3.10;

2. an energy equality; see Lemma 4.3.11;

3. a weak formulation of the pure transport of the interface (4.1.6) in terms of a transport equation for χ; see Lemma 4.3.12 and Deﬁnition 4.3.13.

Based on these three conditions, we introduce a weak formulation of the sharp-interface model (4.1.2)–(4.1.9) in Deﬁnition 4.3.14 below.

Assumptions 4.3.1 (Existence of smooth solutions). In addition to Assumptions 3.2.1, let the following conditions be satisﬁed.

1. Regularity of initial interface. Γ(i) is a C2-hypersurface in the sense of Deﬁni- tion 4.2.1, inducing a disjoint partition Ω=Ω−,(i) Γ(i) Ω+,(i), such that ∪ ∪

Γ(i) = ∂Ω−,(i) Ω and Ω+,(i) =Ω Ω−,(i) =Ω (Ω−,(i) Γ(i)). ⊂⊂ \ \ ∪

Deﬁne the initial associated density function ρ(i) :Ω R by →

(i) ρ (x)= β χ −,(i) (x)+ β χ +,(i) (x) for x Ω (4.3.4) 1 Ω 2 Ω ∈

and deﬁne χ(i) :Ω R by →

(i) χ (x)= χ −,(i) (x) for x Ω. (4.3.5) Ω ∈

2. Regularity of initial velocity. v(i) :Ω Rn belongs to C0(Ω)n. Additionally, the → restrictions v(i) to Ω±,(i) satisfy Ω±,(i)

v(i) C1(Ω±)n and div(v(i)) = 0. Ω±,(i) ∈ 0 Ω±,(i)

3. Existence of smooth solutions. (v,p, Γ) is a solution triplet satisfying equations (4.1.2)–(4.1.9) with the following regularity properties.

74 4.3 The Notion of Weak Solutions

a) Regularity of velocity and pressure. There exist open sets U −, U + Rn+1 ⊂ with Ω±(t) t U ± × { } ⊂ t∈[[0,T ] as well as functions v± C2(U ±)n and p± C1(U ±) such that ∈ ∈ v = v± and p = p± on Ω±(t) t . × { } t∈[[0,T ]

b) Regularity of interface. (Γ(t))t∈[0,T ] is a family of evolving hypersurfaces in the sense of Definition 4.2.10 such that Ω−(t) Γ(t) Ω+(t) is a pairwise disjoint ∪ ∪ partition of Ω and Γ(t) = ∂Ω−(t) Ω for all t [0, T ]. Additionally, for ⊂⊂ ∈ Γ = Γ(t) t , let ν− C0(Γ)n be such that ν−( ,t) is the unit-normal t∈[0,T ] × { } ∈ · fieldS pointing outward to Ω−(t) for all t [0, T ]. ∈ c) Associated density function. Let ρ be defined by (4.3.1).

4.3.1 Solenoidal Functions

In Section 3.6 we already anticipated the use of divergence-free or solenoidal test functions in the weak formulation for the sharp-interface model. This procedure is in the spirit of the theory of the incompressible Navier–Stokes equations; see for example [25, 91]. It leads to a weak formulation lacking the pressure function. As a consequence, one has to reconstruct the pressure from the weak formulation in order to justify this approach. For the free-boundary problem (4.1.2)–(4.1.9), our proceeding will be in the same manner. We therefore briefly recall the definitions and facts about solenoidal functions that we will use in the sequel and introduce the accompanying notation. The test space C∞ (Ω) = ψ C∞(Ω)d : div(ψ) = 0 0,σ ∈ 0 of all smooth divergence-free test functions plays an important role in the treatment of weak solution to the sharp-interface model. Here and in the sequel, the index σ always indicates “divergence free” or “solenoidal“. In the context of Lebesgue and Sobolev spaces, it is convenient to consider the closure of C∞ (Ω) with respect to the Lr- and the W 1,r-norm. For r [1, ], we define 0,σ ∈ ∞

r ∞ k·kLr(Ω)d 1,r ∞ k·kW 1,r(Ω)d Lσ(Ω) = C0,σ(Ω) and W0,σ (Ω) = C0,σ(Ω) .

1 1,2 1,r Furthermore, for r = 2, we use the notation H0,σ(Ω) = W0,σ (Ω). The space W0,σ (Ω) has the following useful characterisation.

Lemma 4.3.2 (Characterisation of W 1,r(Ω)). For d 2 and r (1, ), let Ω Rd be a 0,σ ≥ ∈ ∞ ⊂ bounded domain with Lipschitz boundary. Then there holds

W 1,r(Ω) = u W 1,r(Ω)d : div(u) = 0 . 0,σ ∈ 0

75 Chapter 4 The Sharp-Interface Model

Proof. The lemma is a direct consequence of [91, Lemma II.2.2.3].

For the treatment of non-stationary problems, it is convenient to introduce also the spaces

C∞((0, T ); C∞ (Ω)) = ψ C∞(Ω (0, T ))d : div(ψ) = 0 (4.3.6) 0 0,σ ∈ 0 × and

C∞([0, T ); C∞ (Ω)) = ψ : ψ C∞(Ω ( 1, T ))d, div(ψ) = 0 . (4.3.7) 0 0,σ |Ω×[0,T ) ∈ 0 × − We recall that throughout this thesis the divergence is taken only with respect to the spatial variables. It is customary to use the notation ψ(t)= ψ( ,t) C∞ (Ω) for t (0, T ) · ∈ 0,σ ∈ or [0, T ), respectively. Note that, in general, ψ(0) is non-zero for ψ C∞([0, T ); C∞ (Ω)). ∈ 0 0,σ

4.3.2 Variational Formulation

We seek to derive a variational formulation from equations (4.1.2) and (4.1.3) taking into account conditions (4.1.4)–(4.1.9). As already announced in Section 3.6, to this end, we will use divergence-free test functions. This choice will remove the pressure function from the resulting weak formulation. We postpone the justiﬁcation of this approach to Section 4.5, where we will reconstruct the pressure function from the weak formulation. We start with three remarks on the notation that we will use in the sequel.

Remark 4.3.3 (Time derivatives). Despite the fact that the interface Γ(t) has Lebesgue measure zero, subsequently, we will often use Ω Γ(t)=Ω−(t) Ω+(t) as domain of \ ∪ integration instead of the whole domain Ω. This is because, by Assumptions 4.3.1, the restrictions of ρ and v to the space-time domains

Ω− = Ω−(t) t and Ω+ = Ω+(t) t × { } × { } t∈(0,T ) t∈(0,T ) [ [ are smooth, whereas there is no information about the behaviour of ∂tρ(t) and ∂tv(t) across the interface Γ(t). In particular, we even cannot assume that the time derivatives ∂tρ(t) or ∂tv(t) exist on Γ(t).

Remark 4.3.4 (Spatial derivatives). In view of Assumptions 4.3.1, for any t (0, T ), ∈ spatial derivatives of v(t) and p(t) are smooth in Ω Γ(t). However, jumps across the \ (smooth) interface Γ(t) may occur. Considering these functions on the whole domain Ω, they belong to the space BV (Ω). Their (distributional) derivatives are, roughly speaking, composed by the derivatives in the bulk phases Ω−(t) Ω+(t)=Ω Γ(t) and a jump part ∪ \ across the interface Γ(t). For details we refer to [19, Example 10.2.1]. Below in Proposition 4.3.7, we will compute the distributional gradient of v(t). It will turn out that, in view of the jump condition (4.1.5), v(t) is weakly diﬀerentiable on Ω. This will justify to deal with the (weak) gradient v(t) in the whole domain Ω whenever ∇ Assumptions 4.3.1 are satisﬁed.

76 4.3 The Notion of Weak Solutions

Remark 4.3.5 (Velocity jump across the interface). For ﬁxed t [0, T ], denote the traces ∈ on the interface Γ(t) by upper indices + and . For a given quantity f and x Γ(t), this − ∈ is explicitly

f +(x,t) = lim f(x + ξν−(x,t),t) and f −(x,t) = lim f(x ξν−(x,t),t) ξց0 ξց0 − with respect to Ω+(t) and Ω−(t), respectively. The velocity jump vanishes on Γ(t) due to (4.1.5). Hence v is continuous across the interface. Therefore, we do not need to distinguish between the traces v+( ,t) and v−( ,t), since 0 = [v(t)] = v+(t) v−(t) on · · − Γ(t). In the sequel we will often suppress the upper indices + and and simply write − v(t)= v+(t)= v−(t) on Γ(t), wherever no confusion can arise.

For the treatment of time derivatives in (4.1.2) and (4.1.3) and for later use, we provide the following consequences of the transport theorem (Theorem 4.2.12). These statements will also turn out to be useful in Section 4.3.3, where we shall derive an energy equality for the system (4.1.2)–(4.1.9).

Lemma 4.3.6 (Transport identities). Suppose that Assumptions 4.3.1 are valid. Then, for every t (0, T ) and every ψ C∞ (Ω [0, T ))n, the following statements hold true. ∈ ∈ (0) × d n−1 1. ρv ψ dx = ρ∂t(v ψ)dx + (β β ) V (v ψ)d (x). dt Ω · Ω\Γ(t) · 1 − 2 Γ(t) · H d R 2 R 2 R 2 n−1 2. ρ v dx = ρ∂t v dx + (β β ) V v d (x). dt Ω | | Ω\Γ(t) | | 1 − 2 Γ(t) | | H R R R Proof. For the proof of the ﬁrst statement, we apply Theorem 4.2.12 to obtain

d n−1 v ψ dx = ∂t(v ψ)dx + V (v ψ)d (x) dt − · − · · H ZΩ (t) ZΩ (t) ZΓ(t) and, likewise,

d n−1 v ψ dx = ∂t(v ψ)dx V (v ψ)d (x). dt + · + · − · H ZΩ (t) ZΩ (t) ZΓ(t) Recalling the deﬁnition of ρ from (4.3.1), we infer that d d d ρv ψ dx = β1 v ψ dx + β2 v ψ dx dt · dt − · dt + · ZΩ ZΩ (t) ZΩ (t) n−1 = β1 ∂t(v ψ)dx + V (v ψ)d (x) − · · H ZΩ (t) ZΓ(t) ! n−1 + β2 ∂t(v ψ)dx V (v ψ)d (x) + · − · H ZΩ (t) ZΓ(t) ! n−1 = ρ∂t(v ψ)dx + (β β ) V (v ψ)d (x). · 1 − 2 · H ZΩ\Γ(t) ZΓ(t) The second claim of the lemma now follows in an analogous manner with v taking the role of ψ. This completes the proof.

As already anticipated in Remark 4.3.4, the spatial derivatives of v exist in the weak sense.

77 Chapter 4 The Sharp-Interface Model

Proposition 4.3.7 (Weak differentiability of v). Let t (0, T ). If Assumptions 4.3.1 are ∈ satisfied, then v(t) is weakly differentiable in Ω.

Proof. Let t (0, T ). In view of Assumptions 4.3.1, there holds that ∈ 2 − n 2 + n v(t) − C (Ω (t)) and v(t) + C (Ω (t)) . |Ω (t) ∈ |Ω (t) ∈ Making use of the jump condition (4.1.5), we may compute the distributional derivative of v(t) as follows. For any i = 1,...,n and any ψ C∞(Ω)n, there holds ∈ 0

vi(t) div(ψ)dx = vi(t) div(ψ)dx + vi(t) div(ψ)dx − + ZΩ ZΩ (t) ZΩ (t) − − n−1 = vi(t) ψ dx + vi (t)ψ ν d (x) − − ∇ · · H ZΩ (t) ZΓ(t) + − n−1 vi(t) ψ dx vi (t)ψ ν d (x) − + ∇ · − · H ZΩ (t) ZΓ(t) − n−1 = vi(t) ψ dx [vi(t)]ψ ν d (x). − ∇ · − · H ZΩ\Γ(t) ZΓ(t) Since, by (4.1.5), there holds [vi(t)] = 0, we conclude that

vi(t) div(ψ)dx = vi(t) ψ dx = wi(t) ψ dx, − ∇ · − · ZΩ ZΩ\Γ(t) ZΩ 1 n where wi(t) L (Ω) is given by ∈ loc

vi(t) for x Ω Γ(t), wi(t)= ∇ ∈ \ 0 for x Γ(t).  ∈

Hence vi(t) is weakly diﬀerentiable in Ω with weak gradient vi(t)= wi(t) a.e. in Ω. ∇ The following weak concept of mean curvature will be useful to obtain a variational formulation of (4.1.7).

Lemma 4.3.8 (Weak-mean-curvature functional). Let t (0, T ) and suppose that As- ∈ sumptions 4.3.1 are satisﬁed. For every ψ C1(Ω)n with div(ψ) = 0 in Ω, there holds ∈ κ(t)ν−(t) ψ d n−1(x)= ν−(t) ν−(t) : ψ d n−1(x). (4.3.8) · H ⊗ ∇ H ZΓ(t) ZΓ(t) Proof. Fix t (0, T ) and let ψ C1(Ω)n with div(ψ) = 0 be arbitrary. We apply the ∈ ∈ integration-by-parts formula, see Theorem 4.2.9, to f = ψi and sum over i = 1,...,n. Denoting κ = κ(t), ν− = ν−(t) and Γ=Γ(t), this implies

κν− ψ d n−1(x)= div (ψ)d n−1(x). · H − Γ H ZΓ ZΓ Remarking that div (ψ)= ν− ν− : ψ, as ψ is divergence free, we infer the claim. Γ − ⊗ ∇ Remark 4.3.9. We want to point out that the right-hand side of (4.3.8) is well-defined if Γ is merely the reduced or the essential boundary of a set of finite perimeter. Then one has to interpret ν− as generalised inner (or outer) normal to Γ. Precise definitions of these measure-theoretic concepts are given in Section 2.3.

78 4.3 The Notion of Weak Solutions

Lemma 4.3.10 (Weak form of linear-momentum balance). Let Assumptions 4.3.1 hold true. For every ψ C∞([0, T ); C∞ (Ω)), there holds ∈ 0 0,σ

T ρv ∂tψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt · ⊗ ∇ − Z0 ZΩ T (4.3.9) = ρ(i)v(i) ψ(0) dx 2σ ν− ν− : ψ d n−1(x)dt. − · − st ⊗ ∇ H ZΩ Z0 ZΓ(t)

Proof. Let ψ C∞([0, T ); C∞ (Ω)). Multiplying (4.3.3) by ψ and integrating with respect ∈ 0 0,σ to space and time leads to

T ρ∂tv + ρ(v )v 2µ(ρ) div(Dv)+ p ψ dx dt = 0. (4.3.10) ·∇ − ∇ · Z0 ZΩ\Γ(t) Applying the ﬁrst statement of Lemma 4.3.6 to deal with the time derivative leads to

T T n−1 ρ∂tv ψ dx dt + (β β ) V (v ψ)d (x)dt · 1 − 2 · H Z0 ZΩ\Γ(t) Z0 ZΓ(t) T T d = ρv ∂tψ dx dt + ρv ψ dx dt − · dt · Z0 ZΩ Z0 ZΩ T (i) (i) = ρv ∂tψ dx dt ρ v ψ(0) dx. − · − · Z0 ZΩ ZΩ

To each of the remaining terms in (4.3.10), we shall apply the integration-by-parts formula on the spatial domains Ω−(t) and Ω+(t). Furthermore, we make use of the representation (4.3.2) of the associated density function. For convenience, we will not mention the t- dependence of functions in the remainder of the proof. By (4.1.10), there holds (v )v = div(v v) in Ω Γ(t). This yields ·∇ ⊗ \

ρ((v )v) ψ dx = β1 div(v v) ψ dx + β2 div(v v) ψ dx. ·∇ · − ⊗ · + ⊗ · ZΩ\Γ(t) ZΩ (t) ZΩ (t)

Hence, taking into account (4.1.5), we infer

ρ((v )v) ψ dx ·∇ · ZΩ\Γ(t) − n−1 = β1 v v : ψ dx + (v ν )v ψ d (x) − Ω−(t) ⊗ ∇ Γ(t) · · H ! Z Z (4.3.11) − n−1 + β2 v v : ψ dx (v ν )v ψ d (x) − + ⊗ ∇ − · · H ZΩ (t) ZΓ(t) ! = ρv v : ψ dx + (β β ) (v ν−)v ψ d n−1(x). − ⊗ ∇ 1 − 2 · · H ZΩ ZΓ(t)

Using that v is weakly diﬀerentiable, see Proposition 4.3.7, and Dv : ψ = Dv : Dψ, we ∇

79 Chapter 4 The Sharp-Interface Model

analogously obtain that

µ(ρ) div(Dv) ψ dx · ZΩ\Γ(t)

= µ(β1) div(Dv) ψ dx + µ(β2) div(Dv) ψ dx − · + · ZΩ (t) ZΩ (t) − − n−1 = µ(β1) Dv : ψ dx + µ(β1) ((Dv) ν ) ψ d (x) (4.3.12) − − ∇ · H ZΩ (t) ZΓ(t) + − n−1 µ(β2) Dv : ψ dx µ(β2) ((Dv) ν ) ψ d (x) − + ∇ − · H ZΩ (t) ZΓ(t) = µ(ρ)Dv : Dψ dx [µ(ρ)Dv]ν− ψ d n−1(x). − − · H ZΩ ZΓ(t) In view of div(ψ) = 0, we have

p ψ dx = p div(ψ)dx + p−ν− ψ d n−1(x) ∇ · − − · H ZΩ\Γ(t) ZΩ (t) ZΓ(t) p div(ψ)dx p+ν− ψ d n−1(x) (4.3.13) − + − · H ZΩ (t) ZΓ(t) = [p]ν− ψ d n−1(x). − · H ZΓ(t) Now combining (4.3.10)–(4.3.13) leads to

T (i) (i) 0= ρv ∂tψ ρ v ψ(0) dx − · − · Z0 ZΩ ZΩ T T ρv v : ψ dx dt + 2 µ(ρ)Dv : Dψ dx dt − ⊗ ∇ Z0 ZΩ Z0 ZΩ T (β β ) (V v ν−)v ψ d n−1(x)dt − 1 − 2 − · · H Z0 ZΓ(t) T + 2µ(ρ)Dvν− pν− ψ d n−1(x)dt. − · H Z0 ZΓ(t) We evaluate the integrals over Γ(t) with the help of the interface conditions (4.1.6) and (4.1.7) to conclude

T ρv ∂tψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt · ⊗ ∇ − Z0 ZΩ T = ρ(i)v(i) ψ(0) dx 2σ κν− ψ d n−1(x)dt. − · − st · H ZΩ Z0 ZΓ(t) Finally, we apply Lemma 4.3.8 to the last integral on the right-hand side to obtain the claim (4.3.9).

4.3.3 Energy Equality

In Chapter 3 we derived an energy inequality for the diﬀuse-interface model. This estimate led to a priori estimates suggesting regularity classes for weak solutions. Here, for the sharp-interface model, we will proceed in an analogous manner.

80 4.3 The Notion of Weak Solutions

Lemma 4.3.11 (Energy equality and a priori bounds). Let Assumptions 4.3.1 hold true. For all τ ,τ [0, T ] such that τ τ , the following energy equality is satisﬁed. 1 2 ∈ 1 ≤ 2 τ2 2σ n−1(Γ(τ )) + 1 ρ(τ ) v(τ ) 2 dx + 2 µ(ρ) Dv 2 dx dt stH 2 2 2 | 2 | | | ZΩ Zτ1 ZΩ (4.3.14) = 2σ n−1(Γ(τ )) + 1 ρ(τ ) v(τ ) 2 dx. stH 1 2 1 | 1 | ZΩ Moreover, if the initial energy

2 (i) n−1 (i) 1 (i) (i) E = 2σst (Γ )+ 2 ρ v dx (4.3.15) H Ω Z is ﬁnite, then there holds

v L∞(0, T ; L2 (Ω)) L2(0, T ; H1(Ω)n) and ρ L∞(0, T ; BV (Ω, β , β )). ∈ σ ∩ 0 ∈ { 1 2} Proof. Let τ ,τ [0, T ] be such that τ τ . We proceed similarly to the proof of 1 2 ∈ 1 ≤ 2 Lemma 4.3.10. We multiply (4.3.3) by v and integrate with respect to space and time. This leads to

τ2 ρ∂tv + ρ(v )v 2µ(ρ) div(Dv)+ p v dx dt = 0. (4.3.16) ·∇ − ∇ · Zτ1 ZΩ\Γ(t) We shall evaluate the integral expression successively. For the treatment of the time derivative, we write

τ τ 2 2 2 2 ρ∂tv v dx dt = ρ∂t v dx dt. · | | Zτ1 ZΩ\Γ(t) Zτ1 ZΩ\Γ(t) Application of Lemma 4.3.6 yields

τ2 2 ρ∂tv v dx dt · Zτ1 ZΩ\Γ(t) τ2 d = ρ v 2 dx (β β ) V v 2 d n−1(x) dt dt | | − 1 − 2 | | H Zτ1 ZΩ ZΓ(t) ! (4.3.17) = ρ(τ ) v(τ ) 2 dx ρ(τ ) v(τ ) 2 dx 2 | 2 | − 1 | 1 | ZΩ ZΩ τ2 (β β ) V v 2 d n−1(x)dt. − 1 − 2 | | H Zτ1 ZΓ(t) For the treatment of the remaining terms in (4.3.16), we shall repeatedly integrate by parts with respect to the spatial variable for ﬁxed t (τ ,τ ). Note that, in view of the ∈ 1 2 boundary condition (4.1.8), no boundary integrals over ∂Ω will appear. To save space, we suppress the t-dependence of functions where no ambiguities arise. For the computation of the second term in (4.3.16), we use that v is divergence free on Ω Γ(t) in view of (4.1.4). \ Hence there holds ((v )v) v = div(v v) v = 1 div( v 2 v) in Ω Γ(t). This implies ·∇ · ⊗ · 2 | | \ 2 2 2 ρ((v )v) v dx = β1 div( v v)dx + β2 div( v v)dx Ω\Γ(t) ·∇ · Ω−(t) | | Ω+(t) | | Z Z Z (4.3.18) = (β β ) v 2 v ν− d n−1(x). 1 − 2 | | · H ZΓ(t)

81 Chapter 4 The Sharp-Interface Model

Proceeding as in (4.3.12) and using Dv : v = Dv 2 leads to ∇ | |

µ(ρ) div(Dv) v dx Ω\Γ(t) · Z (4.3.19) = µ(ρ) Dv 2 dx [µ(ρ)Dv]ν− v d n−1(x). − | | − · H ZΩ ZΓ(t) To treat the pressure term in (4.3.16), we may again use that there holds div(v)=0in Ω Γ(t) in view of (4.1.4). Using calculations as in (4.3.13), we infer that \

p v dx = [p]ν− v d n−1(x). (4.3.20) ∇ · − · H ZΩ\Γ(t) ZΓ(t) Now we may combine (4.3.16)–(4.3.20). Altogether, we obtain

1 1 τ2 ρ(τ ) v(τ ) 2 dx ρ(τ ) v(τ ) 2 dx + 2 µ(ρ) Dv 2 dx dt 2 2 | 2 | − 2 1 | 1 | | | ZΩ ZΩ Zτ1 ZΩ 1 τ2 = (β β ) v 2 (V v ν−)d n−1(x)dt 2 1 − 2 | | − · H Zτ1 ZΓ(t) τ2 [2µ(ρ)Dvν− pν−] v d n−1(x)dt − − · H Zτ1 ZΓ(t) τ2 = 2σ κv ν− d n−1(x)dt st · H Zτ1 ZΓ(t) τ2 = 2σ κV d n−1(x)dt, st H Zτ1 ZΓ(t) where we used the interface conditions (4.1.6) and (4.1.7) to see the last two equalities. For the evaluation of the right-hand side, we take into account Theorem 4.2.12 to conclude

τ2 τ2 κV d n−1(x)dt = d n−1(Γ(t)) dt H − dt H Zτ1 ZΓ(t) Zτ1 = n−1(Γ(τ )) n−1(Γ(τ )). H 1 −H 2 Now the last two identities imply the energy inequality (4.3.14). Suppose that the initial energy E(i), see (4.3.15), is ﬁnite and let t [0, T ]. Recall from ∈ (4.3.1) that there holds ρ(t) β , β a.e. in Ω. Making in (4.3.14) the choice τ = 0 and ∈ { 1 2} 1 τ2 = t then implies 1 β v(t) 2 dx E(i). 2 1 | | ≤ ZΩ Hence v L∞(0, T ; L2(Ω)n). Similarly, we obtain that ∈ T T min µ(β ),µ(β ) Dv 2 dx dt µ(ρ) Dv 2 dx dt 1 E(i). { 1 2 } | | ≤ | | ≤ 2 Z0 ZΩ Z0 ZΩ Due to the boundary condition (4.1.8) and using Korn’s inequality [84, Theorem 1.33], we infer that v L2(0, T ; H1(Ω)n). ∈ 0 In the remainder of the proof we ﬁx t (0, T ). As we have div(v(t)) = 0 a.e. in Ω by ∈ (4.1.4), in view of Lemma 4.3.2, we see that v(t) H1 (Ω) L2 (Ω). Hence v belongs to ∈ 0,σ ⊂ σ ∞ 2 L (0, T ; Lσ(Ω)).

82 4.3 The Notion of Weak Solutions

To explore the regularity of ρ, we recall that, in view of (4.3.1), for every t (0, T ), ∈ there holds ρ(t) β , β a.e. in Ω and, in particular, ρ belongs to L∞(0, T ; L∞(Ω)). ∈ { 1 2} Additionally, for the distributional gradient ρ(t), we have ∇

ρ(t), ψ n = ρ(t), div(ψ) h∇ iD(Ω) −h iD(Ω) = ρ(t) div(ψ)dx − ZΩ = β1 div(ψ)dx β2 div(ψ)dx − − − + ZΩ (t) ZΩ (t)

= (β1 β2) div(ψ)dx − − − ZΩ (t) = (β β ) ψ ν−(t)d n−1(x) − 1 − 2 · H ZΓ(t) for all ψ C∞(Ω)n. Consequently, ρ(t) is a ﬁnite Radon measure and there holds ∈ 0 ∇

ρ(t) = sup ρ(t) div(ψ)dx : ψ C1(Ω)n, ψ 1 k∇ kM(Ω) ∈ 0 k k∞ ≤ ZΩ 1 n = (β2 β1)sup div(ψ)dx : ψ C0 (Ω) , ψ ∞ 1 − − ∈ k k ≤ (ZΩ (t) ) = (β β ) (Ω−(t), Ω), 2 − 1 P where the last identity follows from Remark 2.3.8. Due to Assumptions 4.3.1, Ω−(t) has a Lipschitz boundary and Ω−(t) Ω. Then, by Example 2.3.9, we get ⊂⊂

ρ(t) = (β β ) n−1(∂Ω−(t) Ω)=(β β ) n−1(Γ(t)). (4.3.21) k∇ kM(Ω) 2 − 1 H ∩ 2 − 1 H

Finally, from the energy equality (4.3.14), it follows that

sup ρ(t) M(Ω) < . t∈(0,T ) k∇ k ∞

Altogether, we have proven that ρ L∞(0, T ; BV (Ω, β , β )). This ﬁnishes the proof. ∈ { 1 2}

4.3.4 Transport Equation

The interface condition V = v ν−, see (4.1.6), means that the interface is purely trans- · ported by the ﬂuid; see also Remark 4.1.4. This fact can be expressed by the following transport equation for χ, see (4.3.2), in distributional form, cf. for example [1, Section 2.5] and references therein.

Lemma 4.3.12 (Transport equation). Let Assumptions 4.3.1 hold true. Then, for all ϕ C∞ (Ω [0, T )), there holds ∈ (0) × T (i) χ(∂tϕ + v ϕ)dx dt + χ (x)ϕ(0) dx = 0. (4.3.22) ·∇ Z0 ZΩ ZΩ

83 Chapter 4 The Sharp-Interface Model

Proof. Let ϕ C∞ (Ω [0, T )). Applying Theorem 4.2.12 to ϕ and integrating with ∈ (0) × respect to time yields

T T n−1 V ϕ d (x)dt = ∂tϕ dx dt ϕ(0) dx. (4.3.23) H − − − − Z0 ZΓ(t) Z0 ZΩ (t) ZΩ (0) As there holds div(v(t))=0 in Ω−(t) by (4.1.4), we conclude that

T T v ν−ϕ d n−1(x)dt = div(vϕ)dx dt · H − Z0 ZΓ(t) Z0 ZΩ (t) T = v ϕ dx dt. − ·∇ Z0 ZΩ (t) Recalling (4.1.6), we use that V = v ν− on Γ(t) to obtain · T T ∂tϕ dx dt + ϕ(0) dx + v ϕ dx dt = 0. − − − ·∇ Z0 ZΩ (t) ZΩ (0) Z0 ZΩ (t) As χ(t) and χ(i) are the characteristic functions of Ω−(t) and Ω−,(i) =Ω−(0), respectively, see (4.3.2) and (4.3.5), the identity (4.3.22) follows. This ﬁnishes the proof.

The previous result motivates the following deﬁnition.

Deﬁnition 4.3.13 (Weak solutions of the transport equation). For prescribed functions v L2(0, T ; L2 (Ω)) and χ(i) L∞(Ω), χ L∞(Ω (0, T )) is called a weak solution of the ∈ σ ∈ ∈ × transport equation

∂tχ + v χ = 0 in Ω (0, T ), ·∇ × (4.3.24) χ(0) = χ(i) in Ω

provided that for every ϕ C∞ (Ω [0, T )), (4.3.22) holds true. ∈ (0) ×

4.3.5 The Weak Formulation

We seek to introduce a weak formulation for the sharp-interface model (4.1.2)–(4.1.9). To this end, we restrict the class of weak solutions to pairs (ρ, v) satisfying the energy inequality (4.3.14). For well-prepared initial data (ρ(i), v(i)), this suggests the regularity classes ρ L∞(0, T ; BV (Ω, β , β )) and v L∞(0, T ; L2 (Ω)) L2(0, T ; H1(Ω)n). Recalling the ∈ { 1 2} ∈ σ ∩ 0 results of Section 3.5, for a.e. t (0, T ), there exist a measure-theoretic representative set ∈ Ω−(t) Ω and an induced characteristic function χ(t) BV (Ω, 0, 1 ) of ρ(t) such that ⊂ ∈ { } ρ(t) β2 χ(t)= χ − = − . Ω (t) β β 1 − 2 This, in turn, leads to the representation

ρ(t) = (β β )χ − + β = (β β )χ(t)+ β . 1 − 2 Ω (t) 2 1 − 2 2 Notice that this procedure makes the identity

Ω−(t)= x Ω : ρ(t)= β { ∈ 1}

84 4.3 The Notion of Weak Solutions well-deﬁned in a measure-theoretic sense. As Ω−(t) is of bounded variation, we may deﬁne the interface Γ(t) by Γ(t)= ∂∗(Ω−(t)) Ω, ∩ where ∂∗(Ω−(t)) denotes the reduced boundary of Ω−(t). Hence the variational formulation (4.3.9) remains meaningful if we understand the outer unit normal ν− in the (measure- theoretic) sense of the generalised outer unit normal given by

χ − t (Bδ(x)) ν−(x,t)= lim ∇ Ω ( ) for x Γ(t); δ→0 − χ − (Bδ(x)) ∈ ∇ Ω (t)

see Deﬁnition 2.3.10. Additionally, we require χ to solve the corresponding transport equation in the sense of Deﬁnition 4.3.13. As in (4.1.1), we maintain the convenience that Ω−(t) is compactly contained in Ω. Finally, the results of the Lemmas 4.3.10, 4.3.11 and 4.3.12 motivate the following weak formulation of the sharp-interface model (4.1.2)–(4.1.9).

Deﬁnition 4.3.14 (Weak solutions of the sharp-interface model). For β1, β2 > 0 with β < β , let ρ(i), v(i) BV (Ω, β , β ) H1 (Ω) be prescribed initial data, such that 1 2 ∈ { 1 2} × 0,σ the measure-theoretic representative set Ω−(0) of ρ(i) is compactly contained in Ω, i.e., Ω−(0) Ω, and ρ(i) has the representation ⊂⊂

(i) (i) ρ = (β β )χ − + β = (β β )χ + β , 1 − 2 Ω (0) 2 1 − 2 2 where χ(i) is the induced characteristic function of ρ(i) that is given by

ρ(i) β χ(i) = − 2 BV (Ω, 0, 1 ). β β ∈ { } 1 − 2 Then (ρ, v) is called a weak solution of the free-boundary problem (4.1.2)–(4.1.9) with prescribed initial data (ρ(i), v(i)) if the following conditions are fulﬁlled.

1. Regularity of associated density. ρ L∞(0, T ; BV (Ω, β , β )), and the measure- ∈ { 1 2} theoretic representative set Ω−(t) of ρ(t) is compactly contained in Ω; that is, for a.e. t (0, T ), there holds Ω−(t) Ω. ∈ ⊂⊂ 2. Regularity of velocity. v L∞(0, T ; L2 (Ω)) L2(0, T ; H1(Ω)n). ∈ σ ∩ 0 3. Weak form of linear-momentum balance. For each ψ C∞([0, T ); C∞ (Ω)), ∈ 0 0,σ there holds

T (i) (i) ρv ∂tψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt + ρ v ψ(0) dx · ⊗ ∇ − · Z0 ZΩ ZΩ T (4.3.25) = 2σ ν− ν− : ψ d n−1(x)dt, − st ⊗ ∇ H Z0 ZΓ(t) where Γ(t) = ∂∗(Ω−(t)) is the reduced boundary of Ω−(t), and ν−(t) denotes the corresponding generalised outer unit normal.

85 Chapter 4 The Sharp-Interface Model

4. Energy inequality. For a.e. τ [0, T ), including τ = 0, there holds 1 ∈ 1 τ2 2σ n−1(Γ(τ )) + 1 ρ(τ ) v(τ ) 2 dx + 2 µ(ρ) Dv 2 dx dt stH 2 2 2 | 2 | | | ZΩ Zτ1 ZΩ (4.3.26) 2σ n−1(Γ(τ )) + 1 ρ(τ ) v(τ ) 2 dx ≤ stH 1 2 1 | 1 | ZΩ for all τ [τ , T ). 2 ∈ 1

5. Transport equation. The induced characteristic function χ given by χ = χΩ−(·), that is, ρ β χ = − 2 , β β 1 − 2 is a weak solution of the transport equation (4.3.24) with velocity v and prescribed initial data χ(i) in the sense of Deﬁnition 4.3.13.

From now on, we will always consider weak solutions in the sense of the foregoing deﬁnition. For convenience, for any weak solution (ρ, v), we will use the notation

Ω+(t)=Ω (Ω−(t) Γ(t)), \ ∪

where, as in the previous deﬁnition, Ω−(t) denotes the measure-theoretic representative set of ρ(t) and Γ(t)= ∂∗(Ω−(t)). This means that, via Ω = Ω−(t) Γ(t) Ω+(t), this notation ∪ ∪ leads to a pairwise disjoint partition of Ω. Note that if the set Ω−(t) is suﬃciently smooth sets, its topological and reduced boundary coincide, i.e., Γ(t) = ∂∗(Ω−(t)) = ∂(Ω−(t)). This is consistent with Assumptions 4.3.1.

Remark 4.3.15 (Equal-density case). It would also be possible to study the sharp-interface

model in the case of equal densities β1 = β2, cf. [1, 82]. The condition β1 < β2 is only needed to study the sharp-interface limit of the diﬀuse-interface model (3.2.1)–(3.2.6).

Remark 4.3.16 (Energy inequality). The energy inequality (4.3.26) restricts the class of weak solutions to the sharp-interface problem. This approach is in the spirit of the theory of weak solutions for the incompressible Navier–Stokes equations: in this case, for n = 2, weak solutions are unique, whereas, for n = 3, it can be shown that weak solutions are unique if one weak solution satisﬁes an additional regularity assumption, referred to as Serrin’s condition, cf. [91, Theorem V.1.5.1].

Remark 4.3.17 (Lower semi-continuity of the energy functional). For a weak solution (ρ, v) in the sense of Deﬁnition 4.3.14 and t [0, T ), set ∈

2σst 1 2 E(t)= E(ρ(t), v(t)) = ρ(t) M(Ω) + ρ(t) v(t) dx β2−β1 k∇ k 2 | | ZΩ and D(t) = 2 µ(ρ(t)) Dv(t) 2 dx. | | ZΩ

86 4.4 Lebesgue and Sobolev Spaces on Time-Dependent Domains

Then the energy inequality (4.3.26) is equivalent to the statement that, for a.e. τ [0, T ), 1 ∈ including τ1 = 0, there holds

τ2 E(τ )+ D(t)dt E(τ ) 2 ≤ 1 Zτ1 for all τ [τ , T ), cf. (4.3.21). The energy functional 2 ∈ 1

2σst 1 2 (ρ, v) E(ρ, v)= ρ M(Ω) + ρ v dx 7→ β2−β1 k∇ k 2 | | ZΩ is lower semi-continuous on the “natural” function space BV (Ω) L2(Ω)n. Hence, ana- × logously to Remark 3.2.6, we may always assume that the function t E(t) is lower semi- 7→ continuous, and there is no loss of generality in requiring the energy estimate (4.3.26) for all τ [τ , T ) instead of merely for a.e. τ [τ , T ). 2 ∈ 1 2 ∈ 1

4.4 Lebesgue and Sobolev Spaces on Time-Dependent Domains

In this preparatory section we are concerned with the functional-analytic framework for the consistency result. We need to generalise the classical Bochner spaces of functions taking values in Lebesgue or Sobolev spaces. We are interested in functions that take values in Lebesgue or Sobolev spaces on time-dependent domains (Ω(t))t∈[0,T ]. Similar spaces have for instance been introduced in [12, 13, 75, 87].

4.4.1 Transformation to Fixed Domains

We require the family (Ω(t))t∈[0,T ] to be parametrised in the following way, cf. [87, As- sumption 1.1].

Assumptions 4.4.1 (Time evolution). Let Ω Rn, n = 2, 3, be a bounded domain with ⊂ boundary ∂Ω of class C3. Assume that the time evolution of the family (Ω(t)) Ω t∈[0,T ] ⊂ is described via a time-dependent mapping Φ with the subsequent properties.

1. Regularity of initial domain. The initial domain Ω(0) Rn is a bounded domain ⊂ with C3-boundary ∂(Ω(0)).

2. Regularity of evolution. The time evolution of the domain Ω(t) is given by a C3- diﬀeomorphism Φ( ; t): Ω(0) Ω(t)1, i.e., for every t [0, T ], there holds · → ∈ Ω(t)= Φ(ξ; t) : ξ Ω(0) and Ω(t)= Φ(ξ; t) : ξ Ω(0) . ∈ ∈ Denote by ν = ν( ,t) the corresponding outer unit normal and by V = V ( ,t) the normal · · velocity of (∂Ω(t))t∈[0,T ] with respect to ν.

1 In accordance with [87], we use the notation Φ(·; t) instead of Φ(·,t). The semicolon intends to indicate that Φ−1 = Φ−1(x; t) denotes merely the inverse with respect to the space variable x ∈ Ω(t) for t ∈ [0,T ], i.e., there holds Φ(Φ−1(x; t); t)= x and Φ−1(Φ(ξ; t); t)= ξ for x ∈ Ω(t) and ξ ∈ Ω(0).

87 Chapter 4 The Sharp-Interface Model

3. Regularity of Φ. Φ C3,1(Q )n, where Q = Ω(0) (0, T ). ∈ b 0 0 × 4. Preservation of volume. det( Φ(ξ; t)) = 1 for all (ξ,t) Q . ∇ ∈ 0 Remark 4.4.2. By Assumptions 4.4.1, for any t [0, T ], Ω(t) has a C3-boundary ∂(Ω(t)), ∈ and (∂Ω(t))t∈[0,T ] is a family of evolving hypersurfaces in the sense of Deﬁnition 4.2.10.

Remark 4.4.3 (Conservation of volume). We want to stress that, for (suﬃciently regular) solutions of the sharp-interface model (4.1.2)–(4.1.9), the volume of the phases is preserved. Indeed, for instance, for the phase Ω−(t), in view of Theorem 4.2.12, we have

d d Ω−(t) = 1dx = V (t)d n−1(x). dt dt − H ZΩ (t) ZΓ(t)

Due to the incompressibility condition (4.1.4) and the pure transport of the interface (4.1.6), we see that

d Ω−(t) = v(t) ν−(t)d n−1(x)= div(v(t)) dx = 0. dt · H − ZΓ(t) ZΩ (t)

Hence Ω−(t) = Ω−(0) . Below, we will describe the time evolution of the phase Ω−(t) | | | | via a family of isomorphisms (Φ−( ; t)) in the sense of Assumptions 4.4.1. In this · t∈[0,T ] case, from Ω−(t)= Φ−(ξ; t) : ξ Ω−(0) , we infer that ∈ Ω−(t) = 1dx = det( Φ(ξ; t)) dξ = 1dξ = Ω−(0) . − − | ∇ | − ZΩ (t) ZΩ (0) ZΩ (0)

Hence, in Assumptions 4.4.1, the condition det( Φ( ; t)) 1 is adapted to the fact that ∇ · ≡ the volume of the phases is preserved.

Corollary 4.4.4 (Space-time domain). Let Φ be as in Assumptions 4.4.1. Then the function Λ: (ξ,t) (Φ(ξ; t),t) belongs to C3,1(Q )n+1. Moreover, Λ is invertible with 7→ b 0 −1 3,1 n+1 inverse function Λ C (ΩT ) , where ∈ b

n+1 ΩT = Ω(t) t R . × { } ⊂ t∈(0,T ) [ −1 3,1 n In particular, Φ C (ΩT ) and ΩT has a Lipschitz boundary. ∈ b Proof. As Φ C3,1(Q )n by Assumptions 4.4.1, it follows that Λ C3,1(Q )n+1. The ∈ b 0 ∈ b 0 −1 3,1 n+1 implicit function theorem implies that Λ is invertible and Λ C (ΩT ) . Since ∈ b −1 −1 −1 3,1 n Λ (x,t)=(Φ (x; t),t), it follows that Φ C (ΩT ) . Remarking that ∂(Ω (0, T )) ∈ b × is Lipschitz and that ΩT = Λ(Ω (0, T )) ﬁnishes the proof. × Proposition 4.4.5 (Normal velocity). Suppose that Assumptions 4.4.1 hold true. Then, for every (x ,t ) ∂Ω(t) t , there holds 0 0 ∈ t∈[0,T ] × { } S −1 V (x ,t ) = (∂tΦ)(Φ (x ; t ); t ) ν(x ,t ). 0 0 0 0 0 · 0 0

88 4.4 Lebesgue and Sobolev Spaces on Time-Dependent Domains

Proof. For t [0, T ], fix x ∂Ω(t ). By Assumptions 4.4.1, restriction to the respect- 0 ∈ 0 ∈ 0 ive boundaries yields diffeomorphisms Φ−1( ; t ): ∂Ω(t ) ∂Ω(0) and, for t [0, T ], · 0 0 → ∈ Φ( ; t): ∂Ω(0) ∂Ω(t). Therefore, t η(t) = Φ(Φ−1(x ; t ); t) ∂Ω(t) defines a C1- · → 7→ 0 0 ∈ mapping η : [0, T ] Rn with η(t ) = Φ(Φ−1(x ; t ); t ) = x . Thus η is an admissible → 0 0 0 0 0 choice in Definition 4.2.10, which yields

′ −1 V (x ,t )= η (t ) ν(x ,t ) = (∂tΦ)(Φ (x ; t ); t ) ν(x ,t ). 0 0 0 · 0 0 0 0 0 · 0 0

Consequently, V has the stated representation in terms of Φ.

4.4.2 Divergence-Preserving Transformation

By means of the transformation Φ( ; t): Ω(0) Ω(t), we may transform Lebesgue and · → Sobolev functions deﬁned on Ω(t) to functions on Ω(0). For this purpose, for t [0, T ], ∈ we introduce the transformation Φ∗(t) deﬁned by

(Φ (t)f)(ξ) = ( Φ)−1(Φ(ξ; t); t)f(Φ(ξ; t)) (4.4.1) ∗ ∇ for ξ Ω(0) and f : Ω(t) Rn; see [87, equation (10)]. The main properties of the trans- ∈ → formation (4.4.1) are collected in the next lemma; see also [87, Section 3]. In particular, it turns out that Φ∗(t) deﬁnes a divergence-preserving operator.

Lemma 4.4.6 (Properties of Φ∗(t)). Suppose that (Ω(t))t∈[0,T ] is as in Assumptions 4.4.1. Let k = 0, 1, 2, l = 1, 2, q [1, ] and t [0, T ]. Then the operator Φ (t) deﬁned by (4.4.1) ∈ ∞ ∈ ∗ has the following properties.

1. The mapping Φ (t): W k,q(Ω(t))n W k,q(Ω(0))n is an isomorphism. Its inverse oper- ∗ → −1 ator Φ∗ (t) is given by

(Φ−1(t)h)(x) = ( Φ)(x; t)h(Φ−1(x; t)) for h W k,q(Ω(0))n. (4.4.2) ∗ ∇ ∈

2. There are constants C1,C2 > 0, which do not depend on t, such that

C Φ (t)f k,q n f k,q n C Φ (t)f k,q n . (4.4.3) 1k ∗ kW (Ω(0)) ≤ k kW (Ω(t)) ≤ 2k ∗ kW (Ω(0))

3. The mapping Φ (t): W l,q(Ω(t))n W l,q(Ω(0))n is an isomorphism. ∗ 0 → 0 4. If f W 1,q(Ω(t))n, then div(f) Φ( ; t) Lq(Ω(0)) and there holds ∈ ◦ · ∈

div(Φ (t)f) = div(f) Φ( ; t). (4.4.4) ∗ ◦ ·

Moreover, Φ (t): Lq (Ω(t)) Lq (Ω(0)) is an isomorphism. ∗ σ → σ Proof. Throughout this proof let k = 0, 1, 2, l = 1, 2, q [1, ] and t [0, T ] be ﬁxed. ∈ ∞ ∈

89 Chapter 4 The Sharp-Interface Model

1. By Assumptions 4.4.1 and (4.4.1), Φ (t): W k,q(Ω(t))n W k,q(Ω(0))n is a well-deﬁned ∗ → linear map. Next, we check its bijectivity. Let Φ∗(t)f = Φ∗(t)g for some functions f, g W k,q(Ω(t))n. By the deﬁnition of Φ (t), this means ∈ ∗

( Φ)−1(Φ(ξ; t); t)f(Φ(ξ; t)) = ( Φ)−1(Φ(ξ; t); t)g(Φ(ξ; t)) for a.e. ξ Ω(0). ∇ ∇ ∈

Multiplying by ( Φ)(Φ(ξ; t); t) gives f(Φ(ξ; t)) = g(Φ(ξ; t)) for a.e. ξ Ω(0). As ∇ ∈ Φ( ; t): Ω(0) Ω(t) is bijective, we infer that f = g a.e. in Ω(t). Hence Φ (t) is · → ∗ injective on W k,q(Ω(t))n.

For h W k,q(Ω(0))n, due to Assumptions 4.4.1, the function f = ( Φ)( ; t)h(Φ−1( ; t)) ∈ ∇ · · belongs to W k,q(Ω(t))n, and, for a.e. ξ Ω(0), there holds ∈ (Φ (t)f)(ξ) = ( Φ)−1(Φ(ξ; t); t)f(Φ(ξ; t)) ∗ ∇ = ( Φ)−1(Φ(ξ; t); t)( Φ)(Φ(ξ; t); t)h(Φ−1(Φ(ξ; t); t)) ∇ ∇ = h(ξ).

Hence Φ (t) is surjective and Φ−1(t)h = ( Φ)( ; t)h(Φ−1( ; t)). ∗ ∗ ∇ · · 2. We take into account (4.4.1) and that Φ C3,1(Q )n; see Assumptions 4.4.1. With the ∈ b 0 help of the transformation formula, we then conclude that

C Φ (t)f k,q n f k,q n , 1k ∗ kW (Ω(0)) ≤ k kW (Ω(t))

where C1 > 0 is some constant independent of t, because Φ and all spatial derivatives up to order 3 are uniformly bounded in t. As, by Corollary 4.4.4, there holds that −1 3,1 n Φ C (ΩT ) , we may use (4.4.2) and an analogous argumentation to conclude ∈ b that

f k,q n C Φ (t)f k,q n , k kW (Ω(t)) ≤ 2k ∗ kW (Ω(0))

for some constant C2 > 0 independent of t. Hence (4.4.3) is valid.

3. By restriction to ∂Ω(0), Φ( ; t) induces a diﬀeomorphism Φ( ; t): ∂Ω(0) ∂Ω(t). For · · → every f C∞(Ω(t))n, there holds supp(Φ (t)f) Ω(0) by the deﬁnition of Φ (t). In ∈ 0 ∗ ⊂⊂ ∗ l,q n ∞ n l,q n particular, Φ∗(t)f belongs to W0 (Ω(t)) . As C0 (Ω(t)) is dense in W0 (Ω(t)) and l,q n as Φ∗(t) is continuous on W (Ω(t)) , it follows that

Φ (t) W l,q(Ω(t))n W l,q(Ω(0))n. ∗ 0 ⊂ 0 By (4.4.2), there holds supp(Φ−1(t)h) Ω(t) for all h C∞(Ω(0))n. Analogously, ∗ ⊂⊂ ∈ 0 we conclude that Φ−1(t) W l,q(Ω(0))n W l,q(Ω(t))n. ∗ 0 ⊂ 0 This yields that Φ (t): W l,q(Ω(t))n W l,q(Ω(0))n is an isomorphism. ∗ 0 → 0

90 4.4 Lebesgue and Sobolev Spaces on Time-Dependent Domains

4. By direct calculations, (4.4.4) holds true for any f C∞(Ω(t))n W 1,q(Ω(t))n; see for ∈ ∩ example [56, Proposition 2.4]. For an arbitrary function f W 1,q(Ω(t))n, we consider a ∈ ∞ n 1,q n 1,q n sequence (fm)m N C (Ω(t)) W (Ω(t)) with fm f in W (Ω(t)) for m . ∈ ⊂ ∩ → →∞ Then, for every m N, there holds ∈

div(Φ (t)fm) = div(fm) Φ( ; t), (4.4.5) ∗ ◦ · and it remains to pass to the limit m . To treat the left-hand side of (4.4.5), → ∞ we use that div: W 1,q(Ω(0))n Lq(Ω(0)) and Φ (t): W 1,q(Ω(t))n W 1,q(Ω(0))n are → ∗ → continuous linear operators. Therefore, by (4.4.3), there exists a constant C > 0 such that, for any m N, there holds ∈

div(Φ (t)fm) div(Φ (t)f) q = div(Φ (t)(fm f)) q k ∗ − ∗ kL (Ω(0)) k ∗ − kL (Ω(0)) C Φ (t)(fm f) 1,q n ≤ k ∗ − kW (Ω(0)) C fm f 1,q n . ≤ k − kW (Ω(t)) q Consequently, div(Φ (t)fm) div(Φ (t)f) in L (Ω(0)) for m . By the properties ∗ → ∗ →∞ of the divergence operator and the transformation Φ, we similarly conclude that

div(fm) Φ( ; t) div(f) Φ( ; t) q = div(fm f) Φ( ; t) q k ◦ · − ◦ · kL (Ω(0)) k − ◦ · kL (Ω(0)) = div(fm f) q k − kL (Ω(t)) C fm f 1,q n ≤ k − kW (Ω(t)) q for all m N. This means that div(fm) Φ( ; t) div(f) Φ( ; t) in L (Ω(0)) as ∈ ◦ · → ◦ · m . Therefore, sending m in (4.4.5) implies (4.4.4). →∞ →∞ Due to (4.4.4), there holds div(Φ (t)ψ) = div(ψ) Φ( ,t) = 0 in Ω(0) for ψ C∞ (Ω(t)). ∗ ◦ · ∈ 0,σ Since Φ (t)(C∞ (Ω(t))) Φ (t)(W 1,q(Ω(t))n) = W 1,q(Ω(0))n, by Lemma 4.3.2 we see ∗ 0,σ ⊂ ∗ 0 0 that Φ (t)ψ Lq (Ω(0)). Now let f Lq (Ω(t)). By the deﬁnition of Lq (Ω(t)), there ∗ ∈ σ ∈ σ σ ∞ q n exists a sequence (ψm)m N C (Ω(t)) such that ψm f in L (Ω(t)) as m . ∈ ⊂ 0,σ → → ∞ q n q n As Φ (t) is continuous on L (Ω(t)) , there follows Φ (t)ψm Φ (t)f in L (Ω(0)) for ∗ ∗ → ∗ m . By deﬁnition, Lq (Ω(0)) is closed under Lq-convergence which implies that → ∞ σ Φ (t)f Lq (Ω(0))n. Hence Φ (t)(Lq (Ω(t))) Lq (Ω(0)). ∗ ∈ σ ∗ σ ⊂ σ Analogously, we can justify the claim that Φ−1(t)(Lq (Ω(0))) Lq (Ω(t)). Therefore, ∗ σ ⊂ σ Φ (t): Lq (Ω(t)) Lq (Ω(0)) is an isomorphism. ∗ σ → σ This completes the proof.

Remark 4.4.7 (Divergence-preserving transformation). Instead of using the usual pullback operator

(Φ∗(t)f)(ξ)= f(Φ(ξ; t)) (4.4.6) in (4.4.1), we additionally multiplied by ( Φ)−1(Φ(ξ; t); t). Thereby Φ (t) is divergence ∇ ∗ preserving, see (4.4.4), while the pullback operator (4.4.6), in general, does not have this

91 Chapter 4 The Sharp-Interface Model

property. This is adapted to the fact that, in view of (4.1.4), the sharp-interface model deals with divergence-free velocities in each phase, and motivates the deﬁnition of Φ∗(t), cf. also Remark 4.4.3

4.4.3 Functions on Time-Dependent Domains

We shall introduce function spaces of Bochner-type taking values in Lebesgue or Sobolev

spaces on time-dependent domains. For a family (Ω(t))t∈[0,T ] of subdomains of Ω, we are interested in functions of the form t f(t) Lq(Ω(t)) or t f(t) W k,q(Ω(t)). To 7→ ∈ 7→ ∈ define these function spaces, we will always suppose that (Ω(t))t∈[0,T ] satisfies the regularity conditions gathered together in Assumptions 4.4.1. Under these assumptions, for T > 0, the space-time domain n+1 ΩT = (Ω(t) t ) R × { } ⊂ t∈[(0,T ) 1 has a Lipschitz boundary ∂(ΩT ); see Corollary 4.4.4. Now, for functions f L (ΩT ), ∈ loc the distributional derivatives ∂k πf with (k,π) N Nn are well-defined. This allows t ∇ ∈ 0 × 0 us to define the following Bochner-type function spaces.

Deﬁnition 4.4.8 (Lebesgue and Sobolev spaces on time-dependent domains). Suppose that (Ω(t)) satisﬁes Assumptions 4.4.1. Let s, r [1, ] and q N . t∈[0,T ] ∈ ∞ ∈ 0 s r 1 r 1. The space L (0, T ; L (Ω(t))) consists of all f L (ΩT ) such that f(t) L (Ω(t)) for ∈ loc ∈ s a.e. t (0, T ), and (t f(t) r ) L (0, T ). ∈ 7→ k kL (Ω(t)) ∈ 2. The space Ls(0, T ; W q,r(Ω(t))) consists of all f Ls(0, T ; Lr(Ω(t))) such that, for all ∈ π Nn with π q, there holds πf Ls(0, T ; Lr(Ω(t))). ∈ 0 | |∗ ≤ ∇ ∈ 3. The space W 1,s(0, T ; W q,r(Ω(t))) consists of all f Ls(0, T ; W q,r(Ω(t))) such that ∈ s q,r ∂tf L (0, T ; W (Ω(t))). ∈ 4. The vector-valued versions of the above spaces are given by

Ls(0, T ; W q,r(Ω(t))n)= Ls(0, T ; W q,r(Ω(t)))n, W 1,s(0, T ; W q,r(Ω(t))n)= W 1,s(0, T ; W q,r(Ω(t)))n.

5. Let X(t) either stand for W q,r(Ω(t)) or W q,r(Ω(t))n. The space Ls(0, T ; X(t)) is equipped with the norm

1 T s s 0 f(t) X(t) dt if s< , f s = k k ∞ L (0,T ;X(t))  k k essR sup f(t) if s = .  t∈(0,T ) k kX(t) ∞  The space W 1,s(0, T ; X(t)) is equipped with the norm

1 2 2 2 f 1,s = f s + ∂tf s . k kW (0,T ;X(t)) k kL (0,T ;X(t)) k kL (0,T ;X(t))

92 4.4 Lebesgue and Sobolev Spaces on Time-Dependent Domains

Remark 4.4.9. We want to point out that, in the foregoing Deﬁnition 4.4.8 we crucially 1 used that all deﬁned function spaces are subspaces of L (ΩT ).

We may use Φ∗(t) to transform functions from the previous deﬁnitions to functions taking values in time-independent Lebesgue or Sobolev spaces, i.e., functions belonging to the usual Bochner spaces. To this end, we deﬁne Φ∗f by

t Φ (t)f( ,t). (4.4.7) 7→ ∗ ·

−1 Owing to the time-independent bounds on Φ∗(t) and its inverse Φ∗ (t), the transformation properties from Lemma 4.4.6 carry over to Φ∗, as we now show. The function spaces introduced in Deﬁnition 4.4.8 are transformed as follows.

Proposition 4.4.10 (Properties of Φ∗). Suppose that Assumptions 4.4.1 hold true. Let s,q [1, ] and k = 0, 1, 2. Denote by X(τ), τ [0, T ], either of the spaces W k,q(Ω(τ))n, ∈ ∞ ∈ k,q n q W0 (Ω(τ)) or Lσ(Ω(τ)). Then Φ∗, given by (4.4.7), is a diﬀeomorphism between the spaces Ls(0, T ; X(t)) and Ls(0, T ; X(0)) as well as between the spaces

W 1,s(0, T ; Lq(Ω(t))n) Ls(0, T ; W 1,q(Ω(t))n) ∩ and W 1,s(0, T ; Lq(Ω(0))n) Ls(0, T ; W 1,q(Ω(0))n). ∩ s Proof. By Lemma 4.4.6, Φ∗ is an isomorphism between spaces of the form L (0, T ; X(t)) and Ls(0, T ; X(0)). For the proof of the remaining claim, we study the transformation of time derivatives. Let f W 1,s(0, T ; Lq(Ω(t))n) Ls(0, T ; W 1,q(Ω(t))n). Hence Φ f ∈ ∩ ∗ ∈ s 1,q n 1,s q n L (0, T ; W (Ω(0)) ). By the deﬁnition of W (0, T ; L (Ω(t)) ), there holds that ∂tf ∈ s q n s q n L (0, T ; L (Ω(t)) ). To prove that ∂t(Φ f) L (0, T ; L (Ω(0)) ), we use the mapping ∗ ∈ Λ: (ξ,t) (Φ(ξ; t),t), which belongs to C3,1(Ω(0) (0, T ))n+1, by Corollary 4.4.4, and 7→ b × that we may write Φ g = (( Φ)−1 Λ)(g Λ) (4.4.8) ∗ ∇ ◦ ◦ for any g Ls(0, T ; Lq(Ω(0))n). Using the product and the chain rule, we see ∈ −1 ∂t(Φ f)= ∂t (( Φ) Λ)(f Λ) ∗ ∇ ◦ ◦ −1 −1 = ∂t( Φ) Λ (f Λ)+ ( Φ) Λ ∂t(f Λ) ∇ ◦ ◦ ∇ ◦ ◦ −1 −1 = ∂t( Φ) Λ(f Λ)+ ( Φ) Λ ( f Λ)∂tΦ+ ∂tf Λ ∇ ◦ ◦ ∇ ◦ ∇ ◦ ◦ n −1 −1 = ∂t ( Φ) Λ (f Λ)+ ( Φ) Λ ( (∂if Λ)∂tΦi)+ ∂tf Λ . ∇ ◦ ◦ ∇ ◦ i=1 ◦ ◦ X Recalling (4.4.8), it follows that

n −1 ∂t(Φ∗f)= ∂t ( Φ) Λ ( Φ Λ)Φ∗f + Φ∗(∂if)∂tΦi +Φ∗(∂tf . ∇ ◦ ∇ ◦ i=1 X

93 Chapter 4 The Sharp-Interface Model

Since Φ and Λ belong to C3,1(Ω(0) (0, T ))n and C3,1(Ω(0) (0, T ))n+1, respectively, the b × b × −1 functions ∂t ( Φ) Λ , Φ Λ and ∂tΦ are continuous and bounded on Ω(0) (0, T ). ∇ ◦ ∇ ◦ × s 1,q n Moreover, Φ∗f,Φ∗(∂if)and Φ∗(∂tf) belong to L (0, T ; W (Ω(0)) ) for i = 1,...,n. This s q n 1,s q n implies ∂t(Φ f) L (0, T ; L (Ω(0)) ), and thus Φ f W (0, T ; L (Ω(0)) ). ∗ ∈ ∗ ∈ The remaining claim follows by similar arguments as in the proof of Lemma 4.4.6. This concludes the proof.

In the spirit of Theorem 4.2.12, we obtain the following integration-by-parts formula for Sobolev spaces on time-dependent domains.

Lemma 4.4.11 (Integration by parts). Suppose that Assumptions 4.4.1 hold true. For r [1, ), let f W 1,r(0, T ; Lr(Ω(t))) Lr(0, T ; W 1,r(Ω(t))) and let ϕ C∞(Ω (0, T )). ∈ ∞ ∈ ∩ ∈ 0 × Then there holds

T T T n−1 ∂tfϕ dx dt = f∂tϕ dx dt V fϕ d (x)dt. (4.4.9) − − H Z0 ZΩ(t) Z0 ZΩ(t) Z0 Z∂Ω(t)

Proof. By Corollary 4.4.4, the space-time domain ΩT has a Lipschitz boundary. Therefore, ∞ 1,r 1,r C (ΩT ) W (ΩT ) is dense in W (ΩT ); see [39, p. 127, Theorem 3]. As f belongs to ∩ 1,r r r 1,r 1,r W (0, T ; L (Ω(t))) L (0, T ; W (Ω(t)))) = W (ΩT ), there exists an approximating ∩ ∞ 1,r 1,r sequence (fm)m N C (ΩT ) W (ΩT ) such that fm f in W (ΩT ) as m . ∈ ⊂ ∩ → → ∞ Using Theorem 4.2.12, we obtain

T d 0= f ϕ dx dt dt m Z0 ZΩ(t) T T n−1 = ∂t(fmϕ)dx dt + Vfmϕ d (x)dt. H Z0 ZΩ(t) Z0 Z∂Ω(t) Hence T ∂tfmϕ dx dt Z0 ZΩ(t) T T n−1 = fm∂tϕ dx dt Vfmϕ d (x)dt. − − H Z0 ZΩ(t) Z0 Z∂Ω(t) In view of Corollary 4.4.4 and Proposition 4.4.5, the normal velocity V is bounded. Taking into account standard properties of the trace operator [39, p. 133, Theorem 3], we obtain (4.4.9) by letting m in the ﬁnal equation. →∞

4.5 Consistency of the Weak Formulation

In this section we will justify the weak formulation of the sharp-interface model. Under suitable regularity assumptions on the interface Γ and the velocity v, we will show that weak solutions actually satisfy (4.1.2)–(4.1.9) in an appropriate sense of time-dependent Sobolev spaces of the preceding Section 4.4. To this end, we will basically proceed in two steps.

94 4.5 Consistency of the Weak Formulation

1. In the derivation of the weak form of the linear-momentum equation (4.3.25), we removed the pressure function from the weak formulation by means of divergence-free test functions. In Section 4.5.1 we will reconstruct an associated pressure function in the whole space-time domain Ω (0, T ) from the weak formulation. × 2. In Section 4.5.2 we shall readjust the associated pressure function separately in the space-time domains

Ω− = (Ω−(t) t ) and Ω+ = (Ω+(t) t ) × { } × { } t∈[(0,T ) t∈[(0,T ) to satisfy the pressure jump condition (4.1.7) in the normal direction in an appropriate trace sense.

These results will justify the use of divergence-free test functions in the variational formulation (4.3.25). In particular, it will turn out that the jump condition (4.1.7) is meaningful for weak solutions of the sharp-interface model.

4.5.1 Reconstruction of Pressure

The notion of weak solutions for the sharp-interface model incorporates a variational ∞ ∞ formulation using test functions from the space C0 ([0, T ); C0,σ (Ω)); see (4.3.25). Just as in the theory of the incompressible Navier–Stokes equations, this choice removes the pressure function from the weak formulation, cf. [91, Deﬁnition V.1.1.1]. Thus it is not ∞ ∞ clear that the test space C0 ([0, T ); C0,σ (Ω)) is appropriate. To justify this choice, we will prove that, under additional regularity assumptions given below, it is possible to reconstruct a pressure function from the weak formulation.

Assumptions 4.5.1. Suppose that Assumptions 3.2.1 hold true and assume that Ω Rn ⊂ is a bounded domain with boundary ∂Ω of class C3. Let (ρ, v) be a weak solution of the free-boundary problem (4.1.2)–(4.1.9) in the sense of Deﬁnition 4.3.14 with respect to prescribed initial data (ρ(i), v(i)) BV (Ω, β , β ) H1 (Ω), such that the measure- ∈ { 1 2} × 0,σ theoretic representative set Ω−,(i) = Ω−(0) of ρ(i) is compactly contained in Ω and has a C3-boundary. Moreover, let the following regularity properties hold true.

1. Regularity of interface. For any t [0, T ], Φ−( ; t): Ω−(0) Ω−(t) is a diﬀeo- ∈ · → morphism as in Assumptions 4.4.1, such that the time evolution of the measure-theoretic representative set Ω−(t) of ρ(t) is described by Φ−( ; t), i.e., for every t [0, T ], there · ∈ holds

Ω−(t)= Φ−(ξ; t) : ξ Ω−(0) and Ω−(t)= Φ−(ξ; t) : ξ Ω−(0) . ∈ ∈ Additionally, for all t [0, T ], the interface Γ(t)= ∂(Ω−(t)) Ω is compactly contained ∈ ∩ in Ω, that is, Γ(t) = ∂(Ω−(t)) Ω. Denote by ν− = ν−( ,t) the unit normal to ⊂⊂ ·

95 Chapter 4 The Sharp-Interface Model

Γ(t) pointing outward to Ω−(t) and by V = V ( ,t) the normal velocity of (Γ(t)) · t∈[0,T ] with respect to ν−. Similarly, let the time evolution of Ω+(t)=Ω (Ω−(t) Γ(t)) be \ ∪ described by a diﬀeomorphism Φ+( ; t): Ω+(0) Ω+(t) satisfying Assumptions 4.4.1. · → 2. Regularity of velocity. v L2(0, T ; W 2,2(Ω±(t))n) W 1,2(0, T ; L2(Ω±(t))n). ∈ ∩ Remark 4.5.2 (Existence of smooth solutions). The local well-posedness of the free- boundary problem (4.1.2)–(4.1.9) is the content of [59]. Suﬃcient conditions for the existence and uniqueness of classical solutions to (4.1.2)–(4.1.9) are given in [59, Theorem 4]. This result, in particular, ensures the existence of a solution in the spirit of Assump- tions 4.5.1.

4.5.1.1 The Mean-Curvature Functional for Smooth Interfaces

Due to Assumptions 4.5.1, the family of interfaces (Γ(t))t∈[0,T ] has additional regularity properties. This allows us to extend the mean-curvature function to the space-time domain Ω (0, T ). For the proof, we study the transformation of the trace spaces L2(Γ(t)) = × 2 − 1 1 − L (∂(Ω (t))) and H 2 (Γ(t)) = H 2 (∂(Ω (t))).

Lemma 4.5.3 (Transformation of trace spaces). Suppose that Assumptions 4.5.1 hold true and let t [0, T ]. Then the pullback operator Φ− , deﬁned by Φ− u = u Φ−( ,t) for ∈ −t −t ◦ · u L2(Γ(t)), induces linear homeomorphisms ∈ − 2 2 − 1 1 Φ : L (Γ(t)) L (Γ(0)) and Φ : H 2 (Γ(t)) H 2 (Γ(0)), −t → −t → such that − C u 2 Φ u 2 C u 2 (4.5.1) 1k kL (Γ(t)) ≤ k −t kL (Γ(0)) ≤ 2k kL (Γ(t)) for every u L2(Γ(t)), and ∈ − C1 u 1 Φ−tu 1 C2 u 1 k kH 2 (Γ(t)) ≤ k kH 2 (Γ(0)) ≤ k kH 2 (Γ(t))

1 for every u H 2 (Γ(t)) with constants C ,C > 0 independent of u and t. In particular, ∈ 1 2 there are constants C3,C4 > 0 such that

C n−1(Γ(t))) n−1(Γ(0))) C n−1(Γ(t))). (4.5.2) 3H ≤H ≤ 4H Proof. The estimate (4.5.2) follows from (4.5.1) applied to the constant function u 1. ≡ The proof of the remaining claims can be found in [13, Section 5.4.1].

Lemma 4.5.4 (Mean-curvature functional). If Assumptions 4.5.1 hold true, then there exists a function m L∞(0, T ; H1(Ω−(t))) with the following properties. ∈ 1. Let t [0, T ]. For the trace of m(t) on the boundary Γ(t)= ∂Ω−(t), there holds ∈ m(t) = κ(t). (4.5.3) |Γ(t)

96 4.5 Consistency of the Weak Formulation

2. The zero extension K of m to Ω (0, T ) belongs to L∞(0, T ; L2(Ω)n) and satisﬁes ∇ × T T K ψ dx dt = κν− ψ d n−1(x)dt · · H Z0 ZΩ Z0 ZΓ(t) T (4.5.4) = ν− ν− : ψ d n−1(x)dt ⊗ ∇ H Z0 ZΓ(t) for every ψ C∞([0, T ); C∞ (Ω)). ∈ 0 0,σ − 1 1 Proof. Let t [0, T ]. We apply the pullback operator Φ : H 2 (Γ(t)) H 2 (Γ(0)), intro- ∈ −t → duced in the foregoing Lemma 4.5.3, to the mean-curvature function κ(t), which belongs 1 1 to C (Γ(t)) H 2 (Γ(t)) by Assumptions 4.5.1. For notational convenience, we suppress ⊂ − − − the upper index and simply write Φ = Φ and Φ−t =Φ−t in the remainder of this proof. − We deﬁneκ ˜(x,t)=Φ tκ(x,t)= κ(Φ(x; t),t) for x Γ(0) = ∂(Ω (0)). In order to extend − ∈ κ˜(t) to Ω−(0), we solve the Dirichlet problem

∆˜u(t)=0 inΩ−(0), (4.5.5) u˜(t) =κ ˜(t) on Γ(0).

Since, in view of Assumptions 4.5.1, Ω−(0) is a domain with C3-boundary, there exists a weak solutionu ˜ =u ˜(t) H1(Ω−(0)) of (4.5.5) depending on t, which additionally satisﬁes ∈ the estimate − u˜(t) H1(Ω−(0)) C(Ω (0)) κ˜(t) 1 , k k ≤ k kH 2 (Γ(0)) for some constant C(Ω−(0)) > 0, depending on Ω−(0), but independent of t; see [25, Theorem III.4.1]. By Assumptions 4.5.1 and the foregoing Lemma 4.5.3, we infer that for a suitable constant C > 0, independent of t, there holds

− u˜(t) H1(Ω−(0)) C(Ω (0)) κ(t) 1 C. k k ≤ k kH 2 (Γ(t)) ≤

Therefore, the function m:Ω− R defined by m(x,t) =u ˜(Φ−1(x; t),t) for x Ω−(t) → ∈ belongs to L∞(0, T ; H1(Ω−(t))) and, by construction, satisfies (4.5.3). Define K :Ω (0, T ) R, for any (x,t) Ω (0, T ), by × → ∈ ×

m(x,t) if x Ω−(t), K(x,t)= ∇ ∈ 0 if x Ω Ω−(t).  ∈ \  Then K belongs to L∞(0, T ; L2(Ω)n). For every ψ C∞([0, T ); C∞ (Ω)), we then obtain ∈ 0 0,σ T T K(t) ψ(t)dx dt = m(t) ψ(t)dx dt · − ∇ · Z0 ZΩ Z0 ZΩ (t) T = div(m(t)ψ(t)) dx dt − Z0 ZΩ (t) T = m(t)ψ(t) ν−(t)d n−1(x)dt. · H Z0 ZΓ(t)

97 Chapter 4 The Sharp-Interface Model

Taking into account (4.5.3), we conclude that T T K(t) ψ(t)dx dt = κ(t)ψ(t) ν−(t)d n−1(x)dt. · · H Z0 ZΩ Z0 ZΓ(t) Remarking that the last equality in (4.5.4) follows from Lemma 4.3.8 ﬁnishes the proof.

The preceding lemma is crucial in what follows. To explain its importance, we skip the time-independence for a moment. Define : C∞ (Ω) R by M 0,σ → ψ ν− ν− : ψ d n−1(x), 7→ ⊗ ∇ H ZΓ cf. (4.5.4). Lemma 4.5.4 significantly increases the regularity of the functional : by the M definition of a weak solution, without any additional regularity assumption, the interface Γ is the reduced boundary of a set of finite perimeter. Therefore, in general, (only) M 1 ∗ extends to an element of C0 (Ω) , which is far too weak for the application of the ideas we will use in the sequel. Even under additional assumptions, such as Assumptions 4.5.1, making trace theorems for Sobolev functions available, one would at least need ψ W 2,1(Ω)n ∈ to give a meaning to the trace of ψ on Γ and thus make well-defined. In contrast, ∇ M via the representation (4.5.4), extends to an element of L2(Ω)n = (L2(Ω)n)∗. For our M ∼ purposes, it is important to note that, if Assumptions 4.5.1 are satisfied, then Lemma 4.5.4 allows one to replace (4.3.25) by T (i) (i) ρv ∂tψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt + ρ v ψ(0) dx · ⊗ ∇ − · Z0 ZΩ ZΩ T (4.5.6) = 2σ K ψ dx dt − st · Z0 ZΩ for all ψ C∞([0, T ); C∞ (Ω)). We will see below that this characterisation allows us ∈ 0 0,σ to adapt methods from the theory of the incompressible Navier–Stokes equations. Using (4.3.25), this would not be possible, due to the weak regularity properties of the functional , which underlines the importance of Lemma 4.5.4. M In view of (4.5.6), it is convenient to introduce ′(Ω (0, T ))n given by G ∈ D × T , ψ n = ρv ∂tψ + ρv v : ψ 2µ(ρ)Dv : Dψ dx dt hG iD(Ω×(0,T )) · ⊗ ∇ − Z0 ZΩ T (4.5.7) + 2σ K ψ dx dt st · Z0 ZΩ for ψ C∞(Ω (0, T ))n. Note that, due to (4.5.6), for all ψ C∞((0, T ); C∞ (Ω)), there ∈ 0 × ∈ 0 0,σ holds

, ψ n = 0. (4.5.8) hG iD(Ω×(0,T ))

4.5.1.2 Existence of an Associated Pressure Function

We shall prove the existence of an associated pressure function, that is, a distribution p ′(Ω (0, T )) such that ∈ D × ′ n p = ρ∂tv div(ρv v) 2µ(ρ) div(Dv) + 2σ K in (Ω (0, T )) . ∇ − − ⊗ − st D ×

98 4.5 Consistency of the Weak Formulation

The theory of the incompressible Navier–Stokes equations provides us with the following key tool.

Theorem 4.5.5. Let r, s (1, ) and let r′ = r . If Ls(0, T ; W −1,r(Ω)n) satisﬁes ∈ ∞ r−1 F ∈ T ∞ ∞ (t), ψ(t) ,r′ dt = 0 for all ψ C ((0, T ); C (Ω)), W 1 n 0 0,σ hF i 0 (Ω) ∈ Z0 then there exists a unique p Ls(0, T ; Lr(Ω)) satisfying = p in ′(Ω (0, T ))n; that ∈ F ∇ D × is, T T , ψ n = p(t), ψ(t) n dt = p div(ψ)dx dt hF iD(Ω×(0,T )) h∇ iD(Ω) − Z0 Z0 ZΩ for all ψ C∞(Ω (0, T ))n, and, for a.e. t (0, T ), there holds ∈ 0 × ∈ p(t)dx = 0. ZΩ Proof. See [91, Lemma IV.1.4.1].

Although the functional , deﬁned by (4.5.7), vanishes on C∞((0, T ); C∞ (Ω)) due to G 0 0,σ (4.5.8), we cannot directly apply the foregoing theorem to to reconstruct a pressure G function. This is due to a lack of time-regularity. More precisely, the functional T ψ ρv ∂tψ dx dt 7→ · Z0 ZΩ does not belong to any Ls(0, T ; W −1,r(Ω))-space, in general. To circumvent this problem, we improve the properties of this functional by taking into account Assumptions 4.5.1.

Proposition 4.5.6. Suppose that Assumptions 4.5.1 are satisﬁed. Then there holds

T ρ∂tv ψ dx dt · Z0 ZΩ\Γ(t) T T (4.5.9) n−1 = ρv ∂tψ dx dt (β β ) V (v ψ)d (x)dt − · − 1 − 2 · H Z0 ZΩ Z0 ZΓ(t) for any ψ C∞(Ω (0, T ))n. ∈ 0 × Proof. By Assumptions 4.5.1, v belongs to L2(0, T ; H1(Ω±(t))n) W 1,2(0, T ; L2(Ω±(t))n). ∩ This allows one to apply the integration-by-parts formula (Lemma 4.4.11), which yields

T T β1 ∂tv ψ dx dt + β2 ∂tv ψ dx dt − · + · Z0 ZΩ (t) Z0 ZΩ (t) T T n−1 = β1 v ∂tψ dx dt β1 V (v ψ)d (x)dt − − · − · H Z0 ZΩ (t) Z0 ZΓ(t) T T n−1 β2 v ∂tψ dx dt + β2 V (v ψ)d (x)dt − + · · H Z0 ZΩ (t) Z0 ZΓ(t) for any ψ C∞(Ω (0, T ))n. Recalling that ρ = (β β )χ+β ﬁnally yields the claim. ∈ 0 × 1 − 2 2 We now prove some preparatory results, which incorporate the additional properties from Assumptions 4.5.1, before we reconstruct the pressure function with the help of The- orem 4.5.5.

99 Chapter 4 The Sharp-Interface Model

Proposition 4.5.7. If Assumptions 4.5.1 are satisﬁed, then v has the following properties.

1. div(v) = 0 in Ω (0, T ). × 2. For every ϕ C∞(Ω (0, T )), there holds ∈ 0 × T T n−1 V ϕ d (x)dt = χ∂tϕ dx dt H − Z0 ZΓ(t) Z0 ZΩ T (4.5.10) = (v ν−)ϕ d n−1(x)dt. · H Z0 ZΓ(t) 3. For every ψ C∞(Ω (0, T ))n, there holds ∈ 0 × T T V (v ψ)d n−1(x)dt = (v ψ)(v ν−)d n−1(x)dt. (4.5.11) · H · · H Z0 ZΓ(t) Z0 ZΓ(t)

Proof. 1. By Definition 4.3.14, v L∞(0, T ; L2 (Ω)) L2(0, T ; H1(Ω)n). In particular, this ∈ σ ∩ 0 means that v L2(0, T ; H1 (Ω)n). Finally, Lemma 4.3.2 implies the first claim. ∈ 0,σ 2. Let ϕ C∞(Ω (0, T )). The first equality in (4.5.10) follows from the first statement ∈ 0 × of Theorem 4.2.12. For the proof of the second equality in (4.5.10), we use that χ is a weak solution of the transport equation (4.3.24). Thus, by (4.3.22), we have

T T T χ∂tϕ dx dt = χv ϕ dx dt = v ϕ dx dt. − ·∇ − ·∇ Z0 ZΩ Z0 ZΩ Z0 ZΩ (t) Using integration by parts and div(v)=0inΩ (0, T ), it follows × T T T − n−1 χ∂tϕ dx dt = div(v)ϕ dx dt + (v ν )ϕ d (x)dt − − − · H Z0 ZΩ Z0 ZΩ (t) Z0 ZΓ(t) T = (v ν−)ϕ d n−1(x)dt. · H Z0 ZΓ(t) This proves (4.5.10). To justify (4.5.11), we use that, due to Assumptions 4.5.1, there holds v W 1,2(0, T ; L2(Ω−(t))n) L2(0, T ; H1(Ω−(t)))n) = H1(Ω−)n. By Co- ∈ ∩ T rollary 4.4.4, Ω− has a Lipschitz boundary, and therefore C∞(Ω−) H1(Ω−) is dense T T ∩ T 1 − in H (ΩT ); see [39, p. 127, Theorem 3]. This means that there exists an approximating ∞ − n 1 − n 1 − n sequence (vm)m N C (Ω ) H (Ω ) such that vm v in H (Ω ) as m . ∈ ⊂ T ∩ T → T → ∞ For any ψ C∞(Ω (0, T ))n, for m , we obtain ∈ 0 × →∞ 2 − χ ∂t(vm ψ) χ ∂t(v ψ) in L (Ω ) (4.5.12) · · → · · T and 2 − χ (vm ψ) χ (v ψ) in L (Ω ) (4.5.13) ·∇ · → ·∇ · T since χ L∞(Ω (0, T )) and v L2(0, T ; L2(Ω)n). As χ is a weak solution of the ∈ × ∈ transport equation, by (4.3.22), we infer that

T T χ∂t(vm ψ)dx dt = χv (vm ψ)dx dt. · − ·∇ · Z0 ZΩ Z0 ZΩ

100 4.5 Consistency of the Weak Formulation

Now (4.5.12) and (4.5.13) allow us to pass to the limit m . This yields →∞ T T ∂t(v ψ)dx dt = χ∂t(v ψ)dx dt − · · Z0 ZΩ (t) Z0 ZΩ T = χv (v ψ)dx dt. − ·∇ · Z0 ZΩ As the integration-by-parts formula, see Lemma 4.4.11, applies to the left-hand side and as div(v)=0inΩ−, we obtain

T T V (v ψ)d n−1(x)dt = div((v ψ)v)dx dt · H − · Z0 ZΓ(t) Z0 ZΩ (t) T = (v ψ)(v ν−)d n−1(x)dt. · · H Z0 ZΓ(t) This completes the proof.

Next, we explore the regularity of the convective term (v )v. ·∇ Lemma 4.5.8 (Regularity of convective term). Let Assumptions 4.5.1 be satisﬁed. Then (v )v belongs to L2(0, T ; L2(Ω±(t))n). ·∇

Proof. Let Φ∗ be given by (4.4.7). In view of Proposition 4.4.10, there holds that

w =Φ v L2(0, T ; W 2,2(Ω±(0))n) W 1,2(0, T ; L2(Ω±(0))n), (4.5.14) ∗ ∈ ∩ and it is suﬃcient to verify that

2 2 ± w wi L (0, T ; L (Ω (0))) (4.5.15) ·∇ ∈ for i = 1,...,n. To this end, we will use the continuous embedding

(L2(0, T ; H1(Ω±(0))) W 1,2(0, T ; W −1,2(Ω±(0))) ֒ C0([0, T ]; L2(Ω±(0))), (4.5.16 ∩ → cf. (3.3.15). As (4.5.14) implies that

2 1 ± n 1,2 −1,2 ± n w, wi L (0, T ; H (Ω (0)) ) W (0, T ; W (Ω (0)) ), ∇ ∈ ∩ taking into account the embedding (4.5.16), we conclude that

.(w C0([0, T ]; H1(Ω±(0))n) ֒ L∞(0, T ; L6(Ω±(0))n ∈ →

Then, by H¨older’s inequality, we obtain

w wi 2 3 ± w ∞ 6 ± n wi 2 6 ± n , k ·∇ kL (0,T ;L (Ω (0))) ≤ k kL (0,T ;L (Ω (0)) )k∇ kL (0,T ;L (Ω (0)) ) which, in particular, implies (4.5.15). This completes the proof.

101 Chapter 4 The Sharp-Interface Model

Remark 4.5.9 (Time derivatives across the interface). In (4.5.9), the domain of integration is Ω Γ(t)=Ω−(t) Ω+(t) instead of the whole domain Ω, despite the fact that \ ∪ Γ(t) has Lebesgue measure zero. This is because, by Assumptions 4.5.1, the restrictions of v to Ω± belong to some W 1,q(0, T ; Lq(Ω±(t))n)-space. However, this does not give any information about the behaviour of ∂tv on the interface Γ(t). In particular, we cannot assume that ∂tv exists in the sense of weak derivatives on Ω (0, T ). × By means of Proposition 4.5.6, we improve the regularity of the functional ; see (4.5.7). G Proposition 4.5.10. Suppose that Assumptions 4.5.1 hold true and let be as in (4.5.7). G For ψ C∞(Ω (0, T ))n, define ′(Ω (0, T ))n by ∈ 0 × Greg ∈ D × T T , ψ n = ρ∂tv ψ dx dt ρ div(v v) ψ dx dt hGreg iD(Ω×(0,T )) − · − ⊗ · Z0 ZΩ\Γ(t) Z0 ZΩ (4.5.17) T T 2 µ(ρ)Dv : Dψ dx dt + 2σ K ψ dx dt. − st · Z0 ZΩ Z0 ZΩ Then extends to a functional belonging to L2(0, T ; H−1(Ω)n). Moreover, there holds Greg ∞ n , ψ n = , ψ n for all ψ C (Ω (0, T )) . hG iD(Ω×(0,T )) hGreg iD(Ω×(0,T )) ∈ 0 × Proof. In view of Assumptions 4.5.1, Lemma 4.5.8, Definition 4.3.14 and Lemma 4.5.4, extends to a functional belonging to the class L2(0, T ; H−1(Ω)n). Greg Let ψ C∞(Ω (0, T ))n. To prove that and coincide on C∞(Ω (0, T ))n, it suffices ∈ 0 × G Greg 0 × to show that

T ρ(v ∂tψ + v v : ψ)dx dt · ⊗ ∇ Z0 ZΩ T T (4.5.18) = ρ∂tv ψ dx dt ρ div(v v) ψ dx dt. − · − ⊗ · Z0 ZΩ\Γ(t) Z0 ZΩ To this end, we integrate by parts on Ω±(t) and use Proposition 4.5.7 to see

T ρ div ((v v)ψ) dx dt ⊗ Z0 ZΩ T T = β1 div ((v v)ψ) dx dt + β2 div ((v v)ψ) dx dt − ⊗ + ⊗ Z0 ZΩ (t) Z0 ZΩ (t) T = (β β ) (v ν−)(v ψ)d n−1(x)dt 1 − 2 · · H Z0 ZΓ(t) T = (β β ) V (v ψ)d n−1(x)dt. 1 − 2 · H Z0 ZΓ(t) Taking into account Proposition 4.5.6 leads to

T ρ div ((v v)ψ) dx dt ⊗ Z0 ZΩ T T = ρ∂tv ψ dx dt ρv ∂tψ dx dt. − · − · Z0 ZΩ\Γ(t) Z0 ZΩ This implies (4.5.18).

102 4.5 Consistency of the Weak Formulation

Taking into account the additional smoothness Assumptions 4.5.1, we can prove the existence of an associated pressure function.

Theorem 4.5.11 (Reconstruction of associated pressure). Let Assumptions 4.5.1 be sat- isﬁed. Then there exists some function p L2(0, T ; L2(Ω)) such that its restrictions ∈ ± ± 2 1 ± p = p ± to Ω belong to L (0, T ; H (Ω (t))) and satisfy |Ω − − p = β1∂tv + µ(β1)∆v β1(v )v 2σstK a.e. in Ω , ∇ − − ·∇ − (4.5.19) + + p = β ∂tv + µ(β )∆v β (v )v 2σ K a.e. in Ω , ∇ − 2 2 − 2 ·∇ − st where K L2(0, T ; L2(Ω)n) denotes the extension of the mean-curvature function as in ∈ Lemma 4.5.4.

Proof. Let and be as in (4.5.7) and in (4.5.17), respectively. In view of the foregoing G Greg Proposition 4.5.10 and (4.5.8), for any ψ C∞((0, T ); C∞ (Ω)), there holds ∈ 0 0,σ

, ψ n = , ψ n = 0. hGreg iD(Ω×(0,T )) hG iD(Ω×(0,T )) Since L2(0, T ; H−1(Ω)n), Theorem 4.5.5 applies. Hence there exists a function Greg ∈ p L2(0, T ; L2(Ω)) such that, for the distributional gradient p, there holds ∈ ∇ ∞ n p, ψ n = , ψ n for all ψ C (Ω (0, T )) . h∇ iD(Ω×(0,T )) hGreg iD(Ω×(0,T )) ∈ 0 × Concerning the statements about the restriction p− of p to Ω−, we use in the last identity an arbitrary function ψ C∞(Ω−)n. This leads to ∈ 0

p, ψ n h∇ iD(Ω×(0,T )) T = ρ(∂tv + div(v v)) ψ dx dt − − ⊗ · Z0 ZΩ (t) T T 2 µ(ρ)Dv : Dψ dx dt + 2σst K ψ dx dt. − − − · Z0 ZΩ (t) Z0 ZΩ (t) Since ρ = β in Ω− and v L2(0, T ; W 2,2(Ω−(t))n), there follows 1 ∈

p, ψ n h∇ iD(Ω×(0,T )) T = β1(∂tv + div(v v)) ψ dx dt − − ⊗ · Z0 ZΩ (t) T T 2 µ(β1)Dv : Dψ dx dt + 2σst K ψ dx dt − − − · Z0 ZΩ (t) Z0 ZΩ (t) T = β1(∂tv + div(v v)) ψ dx dt − − ⊗ · Z0 ZΩ (t) T T + 2 µ(β1) div(Dv) ψ dx dt + 2σst K ψ dx dt. − · − · Z0 ZΩ (t) Z0 ZΩ (t) 2 2 − n − As ∂tv L (0, T ; L (Ω (t)) ), due to Assumptions 4.5.1, this implies that p belongs to ∈ L2(0, T ; H1(Ω−(t))). Additionally, a.e. in Ω−, there holds

− p = β ∂tv + 2µ(β ) div(Dv) β (v )v + 2σ K ∇ − 1 1 − 1 ·∇ st = β ∂tv + µ(β )∆v β (v )v + 2σ K, − 1 1 − 1 ·∇ st

103 Chapter 4 The Sharp-Interface Model where the last equality follows, since v is divergence free. The statements about p+ follow analogously. This ﬁnishes the proof.

4.5.2 The Pressure Jump

In Theorem 4.5.11 we reconstructed pressure functions in each bulk phase. However, we did not say anything there about the pressure jump. The question left to answer is whether there are pressure functions p± such that the Young–Laplace law (4.1.7) holds true. That is, whether

[p] ν− = (p+ p−)ν− = 2 µ(β )(Dv)+ µ(β )(Dv)− ν− + 2σ κν− (4.5.20) − 2 − 1 st is satisfied on the interface Γ(t). In the remainder of this chapter we will prove that the smoothness Assumptions 4.5.1 allow for an affirmative answer. A first step towards an answer to this question is to understand the ”jump brackets” [ ] · in an appropriate sense: in Theorem 4.5.11 we reconstructed a pressure function p such that, for its restrictions p± to Ω±, there holds

+ 2 1 + − 2 1 − p = p + L (0, T ; H (Ω (t))) and p = p − L (0, T ; H (Ω (t))). |Ω ∈ |Ω ∈ In particular, for a.e. t (0, T ), the traces p±(t) are well-deﬁned in the Sobolev sense, ∈ |Γ(t) and there holds ± 1 p (t) H 2 (Γ(t)). Γ(t) ∈

Therefore, the statement of Theorem 4.5.11 suggests that the pressure jump [p] on the space-time interface Γ= Γ(t) t (4.5.21) × { } t∈(0,T ) [ 2 1 belongs to the space L (0, T ; H 2 (Γ(t))) which is given by either of the equivalent deﬁnitions

2 1 2 1 − L (0, T ; H 2 (Γ(t))) = u : u L (0, T ; H (Ω (t))) |Γ ∈ n o and 2 1 2 1 + L (0, T ; H 2 (Γ(t))) = u : u L (0, T ; H (Ω (t))) . |Γ ∈ n o Likewise, we introduce the n-dimensional version of the latter space and deﬁne

2 1 n 2 1 n L (0, T ; H 2 (Γ(t)) )= L (0, T ; H 2 (Γ(t))) .

To give the jump condition (4.5.20) a meaning, in the remaining part of this chapter, we will interpret the ”jump brackets” [ ] in the sense of Sobolev traces without changing the · notation. More precisely, for a function f L2(0, T ; L2(Ω)) such that the restrictions ∈ ± ± 2 1 ± f = f ± to Ω belong to L (0, T ; H (Ω (t))), we denote |Ω + − [f(t)] = f (t) f (t) − , ∂Ω+(t)∩Ω − ∂Ω (t)

104 4.5 Consistency of the Weak Formulation

+ − ± where f (t) and f (t) − denote the traces of f (t) on the interface Γ(t)= ∂Ω+(t)∩Ω |∂Ω (t) ∂Ω+(t) Ω = ∂Ω−(t) taken with respect to the domains Ω+(t) and Ω−(t), respectively. ∩ 2 2 n ± Analogously, for a function f L (0, T ; L (Ω) ) such that the restrictions f = f ± to ∈ |Ω ± 2 1 ± n Ω belong to L (0, T ; H (Ω (t)) ), we denote [f(t)] = ([fi(t)])i=1,...,n. To construct a pressure function respecting the Young–Laplace law, we provide the following two technical lemmas.

Lemma 4.5.12. Let D Ω be a bounded subdomain of Ω with Lipschitz boundary ∂D ⊂ 1 n and outer normal νD. Then, for a H 2 (∂D) , the following statements are equivalent. ∈

n−1 1. a νD d (x) = 0. ∂D · H R 2. There exists a function u H1(D)n such that ∈

div(u) = 0 in D and u = a on ∂D. (4.5.22)

Proof. Let u H1(D)n satisfy (4.5.22). Then there holds ∈

n−1 n−1 a νD d (x)= u νD d (x)= div(u)dx = 0. · H · H Z∂D Z∂D ZD The opposite direction follows by [43, Theorem IV.1.1].

The following variant of the fundamental lemma of calculus of variations allows one to deal with divergence-free test functions.

2 1 n Lemma 4.5.13. Let Assumptions 4.5.1 hold true. If b L (0, T ; H 2 (Γ(t)) ) satisﬁes ∈ T b(t) ψ(t)d n−1(x)dt = 0 · H Z0 ZΓ(t) for all ψ C∞((0, T ); C∞ (Ω)), then the following statements hold true. ∈ 0 0,σ

− − 1. The tangential projection t τ (b(t)) = b(t) (b(t) ν (t))ν (t) vanishes on Γ(t), i.e., 7→ P − · for a.e. t (0, T ), there holds τ (b(t)) = 0 on Γ(t). ∈ P

− − 2 1 n 2. The normal projection t − (b(t)) = (b(t) ν (t))ν (t) belongs to L (0, T ; H 2 (Γ(t)) ). 7→ Pν · Moreover, for a.e. t (0, T ), there holds ∈

− − (b(t)) = C(t)ν (t) (4.5.23) Pν

on Γ(t), where the function t C(t) belongs to L2(0, T ) and is given by 7→

1 − n−1 C(t)= − b(t) ν (t)d (x) . (4.5.24) Hn 1(Γ(t)) · H ZΓ(t) !

105 Chapter 4 The Sharp-Interface Model

Proof. We split the proof of the lemma into several steps. Step 1. By assumption, for a.e. t (0, T ), the fundamental lemma of the calculus of ∈ variations implies b(t) ψ d n−1(x) = 0 · H ZΓ(t) for all ψ C∞ (Ω). Passing on to H1 (Ω), i.e., the closure of C∞ (Ω) with respect to the ∈ 0,σ 0,σ 0,σ H1-norm, we infer that b(t) u d n−1(x) = 0 (4.5.25) · H ZΓ(t) for all u H1 (Ω) = u H1(Ω)n : div(u) = 0 ; see Lemma 4.3.2. ∈ 0,σ { ∈ 0 } 1 n − n−1 Step 2. For given a H 2 (Γ(t)) with a ν (t)d (x) = 0, we shall construct some ∈ Γ(t) · H function u H1(Ω)n with u = a on Γ(Rt). Applying Lemma 4.5.12 on Ω−(t) and Ω Ω−(t), ∈ 0 \ respectively, there exist functions u H1(Ω−(t))n and u H1(Ω Ω−(t))n such that 1 ∈ 2 ∈ \

− div(u1)=0inΩ (t) and u1 = a on Γ(t) and div(u )=0inΩ Ω−(t), u = a on Γ(t) and u =0 on ∂Ω. 2 \ 2 2 We recall that Γ(t) = ∂Ω−(t) and Γ(t) Ω = by Assumptions 4.5.1. As u and u ∩ ∅ 1 2 coincide on Γ(t) and u2 vanishes on ∂Ω, it follows that the composite function

− u1 in Ω (t), u =  u in Ω Ω−(t)  2 \ 1 n  belongs to H0 (Ω) and satisﬁes div(u) = 0 in Ω and u = a on Γ(t). Using u as test function in (4.5.25) ﬁnally yields

b(t) a d n−1(x)= b(t) u d n−1(x) = 0. (4.5.26) · H · H ZΓ(t) ZΓ(t)

1 n − 1 n Step 3. For a.e. t (0, T ), there holds b(t) H 2 (Γ(t)) and ν (t) C (Γ(t)) by ∈ ∈ ∈ assumption. This means that there are extensions B H1(Ω)n and N − C1(Ω)n such ∈ ∈ that B = b(t) and N − = ν−(t). |Γ(t) Γ(t)

This implies that (B N −)N − belongs to H1(Ω)n . Hence we infer ·

− − − − 1 n (b(t) ν (t))ν (t) = (B N )N H 2 (Γ(t)) · · Γ(t) ∈ and − − 1 n τ (b(t)) = b(t) (b(t) ν (t))ν (t) H 2 (Γ(t)) . P − · ∈ 1 n Step 4. The function τ (b(t)) belongs to H 2 (Γ(t)) and satisﬁes P

− − − − τ (b(t)) ν (t) = (b(t) (b(t) ν (t))ν (t)) ν (t) = 0. P · − · ·

106 4.5 Consistency of the Weak Formulation

In particular, there holds

− n−1 τ (b(t)) ν (t)d (x) = 0. P · H ZΓ(t)

Thus τ (b(t)) is an admissible choice in (4.5.26), which implies P 2 n−1 n−1 τ (b(t)) d (x)= b(t) τ (b(t)) d (x) = 0, |P | H · P H ZΓ(t) ZΓ(t) since

2 − − 2 2 − 2 τ (b) = b (b ν )ν = b (b ν ) |P | − · | | − · − − = b (b (b ν ) ν )= b τ (b). · − · · P n−1 Hence τ (b(t)) = 0 -a.e. on Γ(t), which proves the ﬁrst claim. P H 2 1 n Step 5. From τ (b) = 0, we infer that − (b)= − (b)+ τ (b)= b L (0, T ; H 2 (Γ(t)) ). P Pν Pν P ∈ For the proof of (4.5.23), we consider

1 n − − −1 − a(t)= ν (b(t)) n−1 b(t) ν (t)d (x) ν (t) P − (Γ(t)) Γ(t) · H H Z ! − 1 − n−1 − = b(t) ν (t) − b(t) ν (t)d (x) ν (t). · − Hn 1(Γ(t)) · H ZΓ(t) !

1 n n−1 Then a(t) belongs to H 2 (Γ(t)) and we shall prove that a(t) vanishes -a.e. on Γ(t). H To this end, we use that a(t) is an admissible function in (4.5.26). This follows, since there holds

a(t) ν−(t)d n−1(x) · H ZΓ(t) − 1 − n−1 n−1 = b(t) ν (t) − b(t) ν (t)d (y) d (x) · − Hn 1(Γ(t)) · H H ZΓ(t) ZΓ(t) ! = 0 by the deﬁnition of a(t). Thus we get

b(t) a(t)d n−1(x) = 0 · H ZΓ(t) and, by the deﬁnition of a(t), we conclude that

2 − 2 n−1 1 − n−1 (b(t) ν (t)) d (x)= − b(t) ν (t)d (x) . (4.5.27) · H Hn 1(Γ(t)) · H ZΓ(t) ZΓ(t) ! Moreover, there holds

2 2 − 1 − n−1 a(t) = b(t) ν (t) − b(t) ν (t)d (x) | | · − Hn 1(Γ(t)) · H ZΓ(t) ! − 2 2 − − n−1 = (b(t) ν (t)) n−1 (b(t) ν (t)) b(t) ν (t)d (x) (4.5.28) · − H (Γ(t)) · · H ZΓ(t) 2 1 − n−1 + − b(t) ν (t)d (x) . Hn 1(Γ(t)) · H ZΓ(t) !

107 Chapter 4 The Sharp-Interface Model

Integrating identity (4.5.28) over Γ(t) and taking into account (4.5.27) leads to

a(t) 2 d n−1(x) | | H ZΓ(t) 2 − 2 n−1 1 − n−1 = (b(t) ν (t)) d (x) − b(t) ν (t)d (x) · H − Hn 1(Γ(t)) · H ZΓ(t) ZΓ(t) ! = 0.

Hence a(t) vanishes n−1-a.e. on Γ(t). H Step 6. We shall prove that the function t C(t), given by (4.5.24), is measurable. To this 7→ end, we use the pullback operator Φ− : L2(Γ(t)) L2(Γ(0)) introduced in Lemma 4.5.3 −t → − 2 n and define Bi(t)=Φ bi(t) L (Γ(0)) for i = 1, 2,...,n and t [0, T ]. Then, in −t ◦ ∈ ∈ 2 2 n view of Lemma 4.5.3, the function B = (B1,B2,...,Bn) belongs to L (0, T ; L (Γ(0)) ). In particular, B is Bochner measurable in the sense of Definition 2.1.2. This means that 2 n there exists a sequence (Bm)m N of simple functions Bm : [0, T ] L (Γ(0)) such that, ∈ → 2 n for a.e. t [0, T ], there holds Bm(t) B(t) in L (Γ(0)) as m . For t [0, T ] and ∈ → → ∞ ∈ A L2(Γ(0))n, define ∈

1 − − n−1 (t, A)= − (Φ A) ν (t)d (x) , I Hn 1(Γ(t)) t · H ZΓ(t) ! − − where Φt denotes the inverse of Φ−t. Using again Lemma 4.5.3, we conclude that, for any t [0, T ], (t, ): L2(Γ(0))n R is a linear functional that, for any A L2(Γ(0))n, ∈ I · → ∈ satisﬁes

n−1 − − n−1 (Γ(t)) (t, A) = (Φt A) ν (t)d (x) H |I | Γ(t) · H Z − − Φ A 2 n ν (t) 2 n ≤ k t kL (Γ(t)) k kL (Γ(t)) and, consequently,

n−1 − 1 − (t, A) = (Γ(t)) 2 Φ A 2 n D A 2 n (4.5.29) |I | H k t kL (Γ(t)) ≤ k kL (Γ(0)) for some constant D> 0 independent of t [0, T ]. Hence (t)= (t, ) deﬁnes an element ∈ I I · of L2(Γ(0))n ∗, where the constant of continuity does not depend on t [0, T ]. Then, ∈ Cm: [0, T ] R, deﬁned by Cm(t) = (t,Bm(t)) for t [0, T ], is a simple function for → I ∈ every m N. Moreover, for a.e. t [0, T ], we infer that Cm(t)= (t,Bm(t)) (t,B(t)) ∈ ∈ I →I as m . Since there holds →∞ 1 − − n−1 (t,B(t)) = − (Φ B) ν (t)d (x) I Hn 1(Γ(t)) t · H ZΓ(t) !

1 − n−1 (4.5.30) = − b(t) ν (t)d (x) Hn 1(Γ(t)) · H ZΓ(t) ! = C(t), we conclude that t C(t) is a measurable function. 7→

108 4.5 Consistency of the Weak Formulation

Step 7. Taking into account the identity (4.5.30) as well as the estimate (4.5.29) yields

T T T 2 2 2 C(t) dt = (t,B(t)) dt D B(t) 2 n dt | | |I | ≤ k kL (Γ(0)) Z0 Z0 Z0 for some constant D > 0. Hence we have C 2 D B 2 2 n and, as k kL (0,T ) ≤ k kL (0,T ;L (Γ(0)) ) B L2(0, T ; L2(Γ(0))n), it follows that C L2(0, T ). This ﬁnishes the proof. ∈ ∈ The existence statement of Theorem 4.5.11 and the preparatory Lemma 4.5.13 now allow us to construct a pressure function satisfying the jump condition of the Young–Laplace law.

Theorem 4.5.14 (Reconstruction of pressure). Let Assumptions 4.5.1 be satisﬁed. Then there exists a unique function p L2(0, T ; L2(Ω)) with the following properties. ∈ 2 1 ± 1. p ± L (0, T ; H (Ω (t))). |Ω ∈ 2. For a.e. t (0, T ), there holds p(t)dx = 0. ∈ Ω R − 3. p = β ∂tv + µ(β )∆v β (v )v a.e. in Ω . ∇ − 1 1 − 1 ·∇ + 4. p = β ∂tv + µ(β )∆v β (v )v a.e. in Ω . ∇ − 2 2 − 2 ·∇ − − 2 1 5. [p]=2[µ(ρ)Dvν ] ν + 2σ κ in L (0, T ; H 2 (Γ(t))). · st Proof. We shall construct the desired function p:Ω (0, T ) R with the help of the × → ± functions p from Theorem 4.5.11 and the function K = mχ − from Lemma 4.5.4. To ∇ Ω this end, we proceed in several steps. Step 1. Consider the function

− − p 2σstm in Ω , p˜ =  − + + p in Ω .

 2 1 ± Notice that, by Lemma 4.5.4 and Theorem 4.5.11,p ˜ ± belongs to L (0, T ; H (Ω (t))) |Ω and, in the almost-everywhere sense, there holds

− p˜ = p 2σ m = β ∂tv + µ(β )∆v β (v )v + 2σ K 2σ m ∇ ∇ − st∇ − 1 1 − 1 ·∇ st − st∇ = β ∂tv + µ(β )∆v β (v )v − 1 1 − 1 ·∇ in Ω− and, likewise,

+ p˜ = p = β ∂tv + µ(β )∆v β (v )v + 2σ K ∇ ∇ − 2 2 − 2 ·∇ st = β ∂tv + µ(β )∆v β (v )v − 2 2 − 2 ·∇ a.e. in Ω+. We remark that these properties remain valid for

− − p˜ Ω− + C in Ω , p =  | + + p˜ + + C in Ω  |Ω 

109 Chapter 4 The Sharp-Interface Model for arbitrary functions C−,C+ L2(0, T ). Therefore, it is suﬃcient to prove that there ∈ exists some C L2(0, T ) such that, for a.e. t (0, T ), there holds ∈ ∈

[˜p(t)] = 2[µ(ρ(t))(Dv(t)ν−(t)) ν−(t)] 2σ κ(t)+ C(t) (4.5.31) · − st on Γ(t). This is because the functions C− and C+ provide two degrees of freedom: the first may be used to remove the function C from the previous equation. For example, by making the choice C− = C and C+ = 0, the function p satisfies the desired jump condition. If p does not have the zero-mean property, the second degree of freedom may be used to subtract its mean value. Hence it is sufficient to prove thatp ˜ satisfies (4.5.31).

Step 2. Let ψ C∞((0, T ); C∞ (Ω)) be a divergence-free test function. By the deﬁnition ∈ 0 0,σ ofp ˜, there holds [˜p]=(˜p)+ (˜p)− = p+ (p 2σ m)− = [p] + 2σ κ. This implies − − − st st

T [˜p] ν− ψ d n−1(x)dt · H Z0 ZΓ(t) T T + − = p ψ dx dt (p 2σstm) ψ dx dt − + ∇ · − − ∇ − · Z0 ZΩ (t) Z0 ZΩ (t) T T = p− ψ dx dt p+ ψ dx dt − − ∇ · − + ∇ · Z0 ZΩ (t) Z0 ZΩ (t) T + 2σ κν− ψ d n−1(x)dt. st · H Z0 ZΓ(t)

By Proposition 4.5.7, there holds div(v)=0inΩ (0, T ), and thus (v )v = div(v v) = 0 × ·∇ ⊗ and 2 div(Dv)=∆v in Ω (0, T ). Hence, together with Theorem 4.5.11, there follows ×

− − p = β ∂tv β div(v v) + 2µ(β ) div(Dv) + 2σ K a.e. in Ω , ∇ − 1 − 1 ⊗ 1 st + + p = β ∂tv β div(v v) + 2µ(β ) div(Dv) + 2σ K a.e. in Ω . ∇ − 2 − 2 ⊗ 2 st

We therefore conclude that

T ([˜p] 2σ κ)ν− ψ d n−1(x)dt − st · H Z0 ZΓ(t) T = (β1∂tv + β1 div(v v) 2µ(β1) div(Dv) 2σstK) ψ dx dt − ⊗ − − · Z0 ZΩ (t) T + (β2∂tv + β2 div(v v) 2µ(β2) div(Dv) 2σstK) ψ dx dt + ⊗ − − · Z0 ZΩ (t) T = (ρ∂tv + ρ div(v v) 2µ(ρ) div(Dv) 2σ K) ψ dx dt. ⊗ − − st · Z0 ZΩ\Γ(t)

110 4.5 Consistency of the Weak Formulation

Recalling that v L2(0, T ; H1(Ω)n) and applying integration by parts leads to ∈ T ([˜p] 2σ κ)ν− ψ d n−1(x)dt − st · H Z0 ZΓ(t) T = ρ∂tv ψ dx dt · Z0 ZΩ\Γ(t) T + (ρ div(v v) + 2µ(ρ)Dv : Dψ 2σ K) ψ dx dt ⊗ − st · Z0 ZΩ T + 2 [µ(ρ)Dv] ν− ψ d n−1(x)dt · H Z0 ZΓ(t) T − n−1 = , ψ n + 2 [µ(ρ)Dv] ν ψ d (x)dt, − hGreg iD(Ω×(0,T )) · H Z0 ZΓ(t) where is given by (4.5.17). Since , ψ n = 0, in view of Proposition 4.5.10, Greg hGreg iD(Ω×(0,T )) we conclude that T pν˜ − 2µ(ρ)Dvν− 2σ κν− ψ d n−1(x)dt = 0. − − st · H Z0 ZΓ(t) Finally, by Lemma 4.5.13, there exists a function C L2(0, T ) such that, for a.e. t (0, T ), ∈ ∈ there holds [˜p(t) 2µ(ρ(t))(Dv(t)ν−(t)) ν−(t)] 2σ κ(t) − · − st = [˜p(t)ν−(t) 2µ(ρ(t))Dv(t)ν−(t)] 2σ κ(t)ν−(t) ν−(t) − − st · = C(t) on Γ(t). Hence we conclude (4.5.31). Step 3. It remains to prove the uniqueness of p. Let p ,p L2(0, T ; L2(Ω)) satisfy all the 1 2 ∈ properties of the theorem. For every ψ C∞(Ω (0, T ))n and for i = 1, 2, there holds ∈ 0 × T pi div(ψ)dx dt Z0 ZΩ T T = pi ψ dx dt pi ψ dx dt − − ∇ · − + ∇ · Z0 ZΩ (t) Z0 ZΩ (t) T − n−1 [pi] ν ψ d (x)dt − · H Z0 ZΓ(t) T = β1∂tv µ(β1)∆v + β1(v )v ψ dx dt − − ·∇ · Z0 ZΩ (t) T + β2∂tv µ(β2)∆v + β2(v )v ψ dx dt + − ·∇ · Z0 ZΩ (t) T 2µ(ρ)Dvν− + 2σ κ (ν− ψ)d n−1(x)dt. − st · H Z0 ZΓ(t) Therefore, for any ψ C∞(Ω (0, T ))n, we obtain ∈ 0 × T (p p ) div(ψ)dx dt = 0. 1 − 2 Z0 ZΩ Hence, for a.e. t (0, T ), there exists a constant c = c(t) R such that p (t) p (t)= c(t) ∈ ∈ 1 − 2 a.e. in Ω. Thus, from

p1(t)dx = p2(t)dx = 0, ZΩ ZΩ

111 Chapter 4 The Sharp-Interface Model

we infer that p1(t)= p2(t) a.e. in Ω. This proves the uniqueness of p.

112 Chapter 5

The Sharp-Interface Limit

We now resume the study of the limiting behaviour of weak solutions (ρε, vε) to the diﬀuse- interface model (3.2.1)–(3.2.6) as ε tends to zero. Recalling results from Chapter 3, there holds (at least for suitable subsequences) that vε v and ρε ρ as ε 0 in appropriate → 0 → 0 → function spaces; see Theorems 3.4.1 and 3.4.4. The limiting density function ρ0 belongs to L∞(0, T ; BV (Ω, β , β )), and has the representation { 1 2}

ρ (t) = (β β )χ − + β , 0 1 − 2 Ω (t) 2 with measure-theoretic representative set Ω−(t) Ω, which has ﬁnite perimeter in Ω; see ⊂ Deﬁnition 3.5.4 and Corollary 3.5.6. This, in turn, means that the function

ρ β2 ∞ χ = χ − = − L (0, T ; BV (Ω, 0, 1 )) 0 Ω (·) β β ∈ { } 1 − 2 is an induced characteristic function for ρ0 in the sense of Deﬁnition 3.5.4, and thus encodes all the information of ρ0, cf. Lemma 3.5.3. We are concerned with the sharp-interface limit for the Navier–Stokes–Korteweg system

(3.2.1)–(3.2.6). Roughly speaking, we aim to prove that the pair (ρε, vε) converges to a weak solution (ρ , v ) of the sharp-interface model (4.1.2)–(4.1.9) as ε 0. The results of 0 0 → Chapter 3 already ensure the existence of a convergent subsequence of (ρε, vε)ε∈(0,1) and of a limiting interface. Therefore, it remains to justify the passage to the limit ε 0 in the → weak formulation of (3.2.1)–(3.2.6), and to show that the corresponding limit functions

(ρ0, v0) are indeed weak solutions of (4.1.2)–(4.1.9) in the sense of Deﬁnition 4.3.14. To this end, we will proceed as follows: the required assumptions will be dealt with in Section 5.1. In particular, we will suppose that the energy density

1 ε 2 fε = W (ρε)+ ρε ε 2 |∇ | converges to an appropriate surface measure and that √ρεvε converges to √ρ0v0 in the strong L2-sense as ε tends to zero. These assumptions will be crucial for the analysis of the present chapter. In Section 5.2 we will collect important consequences of the former assumption. In particular, we will control the limiting behaviour of the capillary term.

We will then consecutively verify that (ρ0, v0) is a weak solution of the sharp-interface model (4.1.2)–(4.1.9). To this end, we will, in the conditions that are required by the

113 Chapter 5 The Sharp-Interface Limit

weak formulation (Deﬁnition 3.2.5), pass to the limit ε 0: we will be concerned with → the energy estimate in Section 5.3 and with the variational formulations in Sections 5.4 and 5.5. In Section 5.6 we will eventually put all our results together to perform the sharp-interface limit. In Theorem 5.6.1 we will state the main result of this thesis. We conclude this chapter with a discussion of the convergence assumption for the energy

density fε in Section 5.7.

5.1 Assumptions

We now state suﬃcient conditions for the realisation of the sharp-interface limit. We complement the previous assumptions by appropriate convergence properties of the initial data and of an associated energy functional.

5.1.1 Convergence of Initial Data

We prescribe the convergence of the initial data in the following sense.

Assumptions 5.1.1 (Convergence of initial data). Let Assumptions 3.3.1 be satisﬁed. (i) (i) Additionally, let the initial data (ρε , vε )ε∈(0,1) satisfy the following convergence properties for ε 0. →

1. Constant prescribed mass. There exists some constant m (β Ω , β Ω ) such ∈ 1 | | 2 | | that, for all ε (0, 1), there holds ∈

(i) ρε (x)dx = m. ZΩ

2. Convergence of initial density. For some limit function ρ(i) BV (Ω, β , β ), 0 ∈ { 1 2} (i) (i) p∗ there holds ρε ρ in L (Ω), where p > 2 is as in (3.1.1). Furthermore, the → 0 ∗ measure-theoretic representative set Ω−,(i) Ω has ﬁnite perimeter in Ω; that is, 0 ⊂

(i) (i) ρ0 = (β1 β2)χ0 + β2 = (β1 β2)χ −,(i) + β2 (5.1.1) − − Ω0

(i) (i) and χ0 = χ −,(i) is an induced characteristic function of ρ0 in the sense of Deﬁni- Ω0 tion 3.5.4. The initial interface

Γ(i) = ∂∗(Ω−,(i)) Ω (5.1.2) 0 0 ∩

is compactly contained in Ω, that is, Γ(i) Ω. 0 ⊂⊂

(i) (i) 1 n 3. Convergence of initial velocity. There holds vε v in H (Ω) for some limit → 0 function v(i) H1 (Ω). 0 ∈ 0,σ

114 5.1 Assumptions

4. Convergence of initial energy. The initial energy

2 2 tot,(i) 1 (i) 1 (i) 1 (i) (i) Eε = W (ρε )+ ε ρε + ρε vε dx, Ω ε 2 ∇ 2 Z

see (3.2.7), converges to

2 n−1 (i) 1 (i) (i) 2σst (Γ0 )+ ρ0 v0 dx, H 2 Ω Z

where σst denotes the constant deﬁned in (3.5.2); that is,

β2 σ = min 1 W (z), z a 2 + b2 dz, (5.1.3) st 2 | − | Zβ1 q and W is the double-well potential, cf. (1.4.5) and Assumptions 3.1.1.

Remark 5.1.2 (Volume of initial phases). In view of Assumptions 5.1.1, there holds that

(i) (i) m = ρε (x)dx ρ0 (x)dx = (β1 β2)χ −,(i) + β2 dx → − Ω0 ZΩ ZΩ ZΩ as ε 0. This yields m = (β β ) Ω−,(i) + β Ω . Hence m determines the volume of → 1 − 2 0 2 | | the initial phases via

−,(i) m β2 Ω −,(i) β1 Ω m Ω0 = − | | and Ω Ω0 = | |− . β1 β2 \ β1 β2 − −

Remark 5.1.3. We emphasize that in Assumptions 3.3.1 we only assumed boundedness tot,(i) of the initial energy Eε .

For notational convenience, throughout this chapter we will not relabel subsequences. In particular, without loss of generality, we will assume that the convergence results for

(ρε, vε)ε∈(0,1), proven in Theorems 3.4.1 and 3.4.4, hold true for the entire sequence and not only for an appropriate subsequence.

5.1.2 Convergence of an Associated Energy Functional

Recalling Corollary 3.5.6, for any t [0, T ], the limiting density ρ has the representation ∈ 0

ρ (t) = (β β )χ − + β = (β β )χ (t)+ β (5.1.4) 0 1 − 2 Ω (t) 2 1 − 2 0 2 with measure-theoretic representative set Ω−(t) which has finite perimeter in Ω. Since (5.1.4) is equivalent to ρ0(t) β2 χ (t)= χ − = − , (5.1.5) 0 Ω (t) β β 1 − 2 this allows one to define the limiting interface Γ(t) at time t [0, T ] by ∈ Γ(t)= ∂∗(Ω−(t)) Ω. (5.1.6) ∩ As a consequence of the generalised Gauß–Green theorem (Theorem 2.3.13) and the fact that Ω−(t) is of finite perimeter, the generalised measure-theoretic outer normal ν−(t)

115 Chapter 5 The Sharp-Interface Limit

exists on Γ(t). The set Γ(t) corresponds to the set where ρ0(t) “jumps” in the direction of ν−(t). For simplicity of notation, we denote by Γ the space-time interface

Γ= Γ(t) t . (5.1.7) × { } t∈(0,T ) [ The occurrence of jumps indicates rather weak compactness properties of the sequence

( ρε) . One important idea in Section 3.3.2 was to explore the transformed functions ∇ ε∈(0,1) rε, introduced in (3.3.4), which relate properties of ρε(t) and the energy

1 ε 2 Eε(t)= W (ρε(t)) + ρε(t) dx, (5.1.8) ε 2 |∇ | ZΩ ∞ 1 n ∞ cf. (3.2.8). Both the uniform bounds on rε in L (0, T ; L (Ω) ) and on Eε in L (0, T ), ∇ see Lemma 3.3.6 and cf. (3.2.16), are too weak to identify their corresponding limit functions. We will see that (only) lower bounds are available, via the lower semi-continuity of the variation measure with respect to L1-convergence. However, for our analysis it is an

important point to identify the limit of the energy Eε: the energy equality (3.2.16), to-

gether with Assumptions 3.3.1, implies that the sequence (fε) , for ε (0, 1), defined ε∈(0,1) ∈ by 1 ε 2 fε = W (ρε)+ ρε ε 2 |∇ | ∞ 1 is uniformly bounded in L (0, T ; L (Ω)). Concerning compactness properties of (fε)ε∈(0,1), the problem is that the space L∞(0, T ; L1(Ω)) is not reflexive. Hence boundedness in L∞(0, T ; L1(Ω)) does not in general yield the existence of a convergent subsequence. To circumvent this lack of reflexivity, we shall embed the space L∞(0, T ; L1(Ω)) into an appropriate dual space. To this end, we recall the following definitions and the following representation theorem.

Deﬁnition 5.1.4 (Weakly- measurable functions). Let Y be a Banach space. A func- ∗ tion f : (0, T ) Y ∗ is said to be weakly- measurable if, for any y Y , the function → ∗ ∈ ∞ ∗ t f(t),y Y is measurable. The space L (0, T ; Y ) consists of all weakly- measurable 7→ h i ω∗ ∗ ∗ functions f : (0, T ) Y such that t f(t) Y ∗ is measurable and → 7→ k k

ess sup f(t) Y ∗ < . t∈(0,T ) k k ∞

Theorem 5.1.5 (Representation of weakly- measurable functions). Let Y be a separable ∗ ∞ ∗ 1 ∗ Banach space. Then there holds Lω∗(0, T ; Y ) ∼= L (0, T ; Y ) under the duality T g,f 1 = g(t),f(t) Y dt, h iL (0,T ;Y ) h i Z0 where f L1(0, T ; Y ) and g L∞ (0, T ; Y ∗). ∈ ∈ ω∗ Proof. See [77, Theorem 6.14].

116 5.2 Capillary Term

Due to Theorems 5.1.5 and 2.2.7, there holds

.∗((L∞(0, T ; L1(Ω)) ֒ L∞ (0, T ; (Ω)) = L1(0, T ; C (Ω → ω∗ M ∼ 0

The continuity of this embedding allows us to extract a subsequence of (fε)ε∈(0,1) converging weakly- in L1(0, T ; C (Ω))∗. We shall study the situation where we can identify the ∗ 0 limit function as follows.

Assumptions 5.1.6 (Convergence of associated energy functional). Suppose that Assump- tions 5.1.1 hold true. Let Γ be the limiting space-time interface as in (5.1.7) and let σst be the constant given by (5.1.3). Suppose that, for any t [0, T ], Γ(t) is compactly contained ∈ in Ω; that is, Γ(t) Ω. Furthermore, assume that, as ε 0, there holds ⊂⊂ →

∗ n−1 1 1 ∗ fε ⇁f = 2σ ( )xΓ weakly- in L (0, T ; C (Ω)) , st H ⊗L ∗ 0 where n−1 1 denotes the product measure of n−1 and 1. This means that, for every H ⊗L H L ϕ L1(0, T ; C (Ω)), as ε 0, there holds ∈ 0 → T T 1 ε 2 n−1 ε W (ρε)+ 2 ρε ϕ dx dt 2σst ϕ d (x)dt. (5.1.9) 0 Ω |∇ | → 0 Γ(t) H Z Z Z Z We will perform the sharp-interface limit under Assumptions 5.1.1 and 5.1.6 and postpone the discussion of these additional assumptions to Section 5.7, where, in particular, in Section 5.7.2 we will present special situations where Assumptions 5.1.6 are satisﬁed.

5.2 Capillary Term

The main task in passing to the limit ε 0 in the variational formulation (3.2.15) is → controlling the convergence of the capillary term

T ε ρε ρε : ψ dx dt; ∇ ⊗∇ ∇ Z0 ZΩ see also the discussion in Section 3.6. For this purpose, the following lemma is crucial.

Lemma 5.2.1. Let 0 τ <τ T . For Eε as in (5.1.8) and σ as in (5.1.3), Assump- ≤ 1 2 ≤ st tions 5.1.6 imply the following statements.

1. Convergence of total mass. Let (rε)ε∈(0,1) be as in (3.3.4). Then there holds

τ2 τ2 τ2 1 n−1 lim rε(t) dx dt = lim Eε(t)dt = σst (Γ(t)) dt. (5.2.1) ε→0 |∇ | 2 ε→0 H Zτ1 ZΩ Zτ1 Zτ1

2. Equipartition of energy.

τ2 1 1 2 lim ε W (ρε) 2 ε ρε dx dt = 0. (5.2.2) ε→0 τ Ω − |∇ | Z 1 Z

117 Chapter 5 The Sharp-Interface Limit

In particular, there holds

τ2 τ2 τ2 1 1 2 n−1 lim W (ρε)dx dt = lim ε ρε dx dt = σst (Γ(t)) dt. ε→0 ε ε→0 2 |∇ | H Zτ1 ZΩ Zτ1 ZΩ Zτ1 Proof. 1. Let χ0 = χΩ−(·) be as in (5.1.5). On the one hand, by Corollaries 3.4.3 and 3.5.6 and Fatou’s lemma, we have

τ2 τ2 τ2 σst χ0(t) (Ω)dt = r0(t) (Ω)dt lim inf rε(t) dx dt, |∇ | |∇ | ≤ ε→0 |∇ | Zτ1 Zτ1 Zτ1 ZΩ where χ (t) (Ω) and r (t) (Ω) denote the total variation of χ (t) and ρ (t), |∇ 0 | |∇ 0 | ∇ 0 ∇ 0 respectively, in Ω, cf. Deﬁnition 2.2.4. On the other hand, using the Modica–Mortola trick (3.3.13), we obtain

τ2 τ2 τ2 1 n−1 lim sup rε(t) dx dt lim sup Eε(t)dt = σ (Γ(t)) dt, |∇ | ≤ 2 st H ε→0 Zτ1 ZΩ ε→0 Zτ1 Zτ1

where the last equality follows as ϕ(x,t)= χ(τ1,τ2)(t) is admissible in (5.1.9). Since by Theorem 2.3.13 and Remark 2.3.14, for a.e. t (0, T ), we have ∈ n−1 ∗ − n−1 χ (t) (Ω) = χ − (Ω) = (∂ Ω (t) Ω) = (Γ(t)), |∇ 0 | ∇ Ω (t) H ∩ H

this establishes (5.2.1).

2. We use the argumentation of [69, Lemma 1]: for ε (0, 1), deﬁne ∈ 1 1 aε = ε ρε and bε = W (ρε). 2 |∇ | ε q q From the energy estimate (3.2.16) and Assumptions 3.3.1, we infer that (aε)ε∈(0,1) and ∞ 2 (bε)ε∈(0,1) are uniformly bounded in L (0, T ; L (Ω)). Therefore, by H¨older’s inequality, for some constant C > 0, there holds T 2 2 aε bε dx dt C aε bε L2(0,T ;L2(Ω)). 0 Ω − ≤ k − k Z Z

What remains is to show that the right-hand side of the latter estimate vanishes if 2 ε 2 1 ε 0. We have (aε bε) = ρε + W (ρε) √2 ρε W (ρε). Thus, by the → − 2 |∇ | ε − |∇ | deﬁnition of rε, see (3.3.4), we conclude that p T T T 2 (aε bε) dx dt Eε(t)dt 2 rε(t) dx dt. − ≤ − |∇ | Z0 ZΩ Z0 Z0 ZΩ Finally, (5.2.1) yields the second claim of the lemma. The remaining statement follows directly from (5.2.1) and (5.2.2).

In order to explore the convergence of the capillary term with the help of the continuity theorem of Reshetnyak, see Theorem 2.2.13, we will consider the convergence properties of rε established in previous Lemma from a measure-theoretic viewpoint and study the ∇ n+1 limiting behaviour of the induced Radon measure λε = rε given by ∇ L n+1 λε(B)= rε d for B (Ω (0, T )). (5.2.3) ∇ L ∈B × ZB To simplify the presentation, we do not indicate restrictions to the space-time cylinder Ω (0, T ) and we simply write n+1 instead of the restriction measure n+1x(Ω (0, T )). × L L ×

118 5.2 Capillary Term

Proposition 5.2.2 (Total variation and Radon–Nikodym derivative of λε). Suppose that

Assumptions 5.1.1 and 5.1.6 hold true. For ε (0, 1), let rε be as in (3.3.4). Then λε ∈ n deﬁned by (5.2.3) belongs to (Ω (0, T )) . Moreover, the total variation λε of λε is M × | | given by n+1 n+1 λε (B)= rε (B)= rε d (5.2.4) | | |∇ |L |∇ | L ZB λε and for the Radon–Nikodym derivative of λε with respect to λε , there holds |λε| | |

λε rε (x,t)= ∇ (x,t) (5.2.5) λε rε | | |∇ | for λε -a.e. (x,t) Ω (0, T ). | | ∈ × ∞ 1 n Proof. Let ε (0.1). Recall from Lemma 3.3.6 that rε L (0, T ; L (Ω)) . By con- ∈ ∇ ∈ n struction, λε belongs to (Ω (0, T )) . M × From the deﬁnition of the total variation, we see that (5.2.4) holds true. Details can for example be found in [16, Proposition 1.23]. For any B (Ω (0, T )), we have ∈B × n+1 rε n+1 rε λε(B)= rε d = ∇ rε d = ∇ d λε . B ∇ L B rε · |∇ | L B rε | | Z Z |∇ | Z |∇ | From Corollary 2.2.12, we then infer that (5.2.5) is valid, which completes the proof.

Next, we explore the convergence of the Radon measure λε as ε 0. →

Proposition 5.2.3 (Convergence of λε). Let Assumptions 5.1.1 and 5.1.6 be satisfied. Let n − n−1 1 λε be given by (5.2.3). Define λ (Ω (0, T )) by λ = σ ν xΓ, where 0 ∈ M × 0 st H ⊗L n−1 1 denotes the product measure of the (n 1)-dimensional Hausdorff measure H ⊗L − n−1 and the one-dimensional Lebesgue measure 1 and Γ is the space-time interface; see H L (5.1.7). That is, for any B (Ω (0, T )), the Radon measure λ is given by ∈B × 0 λ (B)= σ ν− d( n−1 1) (5.2.6) 0 st H ⊗L ZB∩Γ for B (Ω (0, T )). Then the total variation of λ is given by ∈B × 0 λ = σ ( n−1 1)xΓ | 0| st H ⊗L and the Radon–Nikodym derivative of λ with respect to λ is given by 0 | 0| λ 0 (x,t)= ν−(x,t) λ | 0| for λ -a.e. (x,t) Ω (0, T ). Moreover, as ε 0, there holds | 0| ∈ × → ∗ n λε ⇁ λ weakly- in (Ω (0, T )) (5.2.7) 0 ∗ M × and

lim λε (Ω (0, T )) = λ0 (Ω (0, T )). (5.2.8) ε→0 | | × | | ×

119 Chapter 5 The Sharp-Interface Limit

λ0 Proof. The total variation λ0 and the Radon–Nikodym derivative of can be obtained | | |λ0| as in Proposition 5.2.2.

To explore the convergence of λε, we recall from Lemma 3.3.6 that (rε)ε∈(0,1) is a uniformly ∞ 1,1 bounded sequence in L (0, T ; W (Ω)). In particular, ( rε) is uniformly bounded ∇ ε∈(0,1) in L1(0, T ; L1(Ω))n. For every η C1(Ω (0, T ))n with compact support in Ω (0, T ), ∈ × × we therefore obtain

T T rε η dx dt = rε div(η)dx dt. (5.2.9) ∇ · − Z0 ZΩ Z0 ZΩ

Since, by Corollaries 3.4.3 and 3.5.6, there holds rε r = σ χ − + Φ(β ) in → 0 − st Ω (·) 2 L∞(0, T ; L1(Ω)) for ε 0, we obtain → T T T lim rε η dx dt = σst div(η)dx dt Φ(β2) div(η)dx dt ε→0 ∇ · − − Z0 ZΩ Z0 ZΩ (t) Z0 ZΩ T = σ η ν− d n−1(x)dt, st · H Z0 ZΓ(t)

where we used the generalised Gauß–Green formula; see Theorem 2.3.13. As rε is uni- ∇ formly bounded in L∞(0, T ; L1(Ω)n) and by approximation, for every η C (Ω (0, T ))n, ∈ 0 × we have T T − n−1 lim rε η dx dt = σst η ν d (x)dt, ε→0 ∇ · · H Z0 ZΩ Z0 ZΓ(t) which means that

lim η dλε = η dλ0. ε→0 ZΩ×(0,T ) ZΩ×(0,T ) ∗ n Hence there holds λε ⇁ λ weakly- in (Ω (0, T )) as ε 0. Moreover, in view of 0 ∗ M × → Lemma 5.2.1 and the deﬁnition of λ0, there holds

T T n−1 λε (Ω (0, T )) = rε dx dt σ (Γ(t)) dt = λ (Ω (0, T )) | | × |∇ | → st H | 0| × Z0 ZΩ Z0 for ε 0. This completes the proof. →

We are now in a position to control the capillary term as ε 0. → Theorem 5.2.4 (Convergence of capillary term). Let Assumptions 5.1.1 and 5.1.6 be satisﬁed. Then, for every ψ C∞([0, T ); C∞ (Ω)), there holds ∈ 0 0,σ T T − − n−1 lim ε ρε ρε : ψ dx dt = 2σst ν ν : ψ d (x)dt, (5.2.10) ε→0 ∇ ⊗∇ ∇ ⊗ ∇ H Z0 ZΩ Z0 ZΓ(t) where Γ(t) is deﬁned by (5.1.6) and ν− denotes the generalised outer normal in the measure- theoretic sense.

Proof. We adapt the proof of [69, Lemma 2] to our situation. Let ψ C∞([0, T ); C∞ (Ω)). ∈ 0 0,σ We express the left-hand side of (5.2.10) in terms of rε deﬁned in (3.3.4): for t (0, T ), ∈

120 5.3 Energy Inequality

∇ρε(t) ∇rε(t) let Ωε(t)= ρε(t) = 0 = rε(t) = 0 . The deﬁnition of rε implies = in {∇ 6 } {∇ 6 } |∇ρε(t)| |∇rε(t)| Ωε(t) and, due to Lemma 5.2.1, we conclude that

T lim ε ρε ρε : ψ dx dt ε→0 ∇ ⊗∇ ∇ Z0 ZΩ T ∇ρε ∇ρε 2 = lim ε : ψ ρε dx dt ε→0 |∇ρε| ⊗ |∇ρε| ∇ |∇ | Z0 ZΩε(t) T = 2 lim ∇rε ∇rε : ψ r dx dt, |∇rε| |∇rε| ε ε→0 0 Ωε(t) ⊗ ∇ |∇ | Z Z provided one of the limits exists. Now let λε be as in (5.2.3). Taking into account Propos- ition 5.2.2, we obtain

T lim ε ρε ρε : ψ dx dt ε→0 ∇ ⊗∇ ∇ Z0 ZΩ T (5.2.11) λε = 2 lim (x,t), (x,t) d λε (x,t), ε→0 F |λε| | | Z0 ZΩ where (y,τ),q for (y,τ),q Ω (0, T ) Sn−1 is given by F ∈ × × q q : ψ(y,τ) if q = 0, (y,τ),q =  ⊗ ∇ 6 F 0 if q = 0.  To explore the convergence of the right-hand side of (5.2.11), we use Propositions 5.2.2 and 5.2.3 and apply Reshetnyak’s continuity theorem (Theorem 2.2.13). Then we ﬁnally conclude that T T λ0 lim ε ρε ρε : ψ dx dt = 2 (x,t), (x,t) d λ0 (x,t) ε→0 ∇ ⊗∇ ∇ F |λ0| | | Z0 ZΩ Z0 ZΩ T = 2σ ν− ν− : ψ d n−1(x)dt, st ⊗ ∇ H Z0 ZΓ(t) which is the desired conclusion.

5.3 Energy Inequality

The next step towards the sharp-interface limit is to let ε 0 in the energy equal- → tot ity (3.2.16). We shall suppose that the kinetic part of the energy Eε , see (3.2.9), has the following asymptotic behaviour as ε 0. → Assumptions 5.3.1. Let Assumptions 3.3.1 hold true. Additionally, suppose that, as ε 0, there holds → 2 2 n √ρεvε √ρ v in L (0, T ; L (Ω) ). (5.3.1) → 0 0

We aim to show that the pair (ρ0, v0) satisﬁes the energy inequality (4.3.26). This is partly done in the following corollary, which is a direct consequence of Assumptions 5.3.1. Subsequently, we shall combine this result with the convergence properties of the energy

Eε from Lemma 5.2.1.

121 Chapter 5 The Sharp-Interface Limit

Corollary 5.3.2. If Assumptions 5.3.1 hold true, then, for every sequence (ρεj , vεj )j∈N, there exists a subsequence (ρε , vε )m N such that, as m , there holds jm jm ∈ →∞

2 ∗ 2 ∞ 1 ∗ ρεjm vεjm dx ⇁ ρ0 v0 dx weakly- in L (0, T ) = L (0, T ) . (5.3.2) Ω Ω | | ∗ ∼ Z Z

2 ∞ Proof. The functions t ρε(t) vε(t) dx are uniformly bounded in L (0, T ) with 7→ Ω | | respect to ε (0, 1), becauseR of the energy inequality (3.2.16) and Assumptions 3.3.1. ∈ Thus, after passing to a suitable subsequence, we obtain

2 ∗ ∞ 2 ( ρε vε dx ⇁ g weakly- in L (0, T ) ֒ L (0, T | | ∗ → ZΩ for some g L∞(0, T ). This yields ∈

2 2 1 .( ρε vε dx ⇀ g weakly in L (0, T ) ֒ L (0, T | | → ZΩ In view of (5.3.1), we may identify g = ρ v 2 dx, which ﬁnishes the proof. Ω 0 | 0| R We provide a technical lemma due to Abels [3], which is useful for passing to the limit in the energy inequality (3.2.16).

Lemma 5.3.3. Let E : [0, T ) [0, ) be a lower semi-continuous function and let → ∞ D : (0, T ) [0, ) be an integrable function. Then the following statements are equi- → ∞ valent.

1. For all τ W 1,1(0, T ) with τ 0 and τ(T ) = 0, there holds ∈ ≥ T T E(0)τ(0) + E(t)τ ′(t)dt D(t)τ(t)dt. ≥ Z0 Z0 2. For all τ τ < T and a.e. 0 τ < T , including τ = 0, there holds 1 ≤ 2 ≤ 1 1 τ2 E(τ )+ D(t)dt E(τ ). 2 ≤ 1 Zτ1 Proof. See [3, Lemma 4.3].

The preliminary work in Lemma 5.2.1, Corollary 5.3.2 and Lemma 5.3.3 allows us to pass to the limit ε 0 in the energy equality (3.2.16). → Theorem 5.3.4 (Energy inequality). Let Assumptions 5.1.1 and 5.1.6 be satisﬁed. Then the inequality

τ2 2σ n−1(Γ(τ )) + 1 ρ (τ ) v (τ ) 2 dx + 2 µ(ρ ) Dv 2 dx dt stH 2 2 0 2 | 0 2 | 0 | 0| ZΩ Zτ1 ZΩ (5.3.3) 2σ n−1(Γ(τ )) + 1 ρ (τ ) v (τ ) 2 dx ≤ stH 1 2 0 1 | 0 1 | ZΩ holds for all τ τ < T and almost all 0 τ < T , including τ = 0. 1 ≤ 2 ≤ 1 1

122 5.3 Energy Inequality

Proof. Let ε (0, 1). By Definition 3.2.5, any weak solution (ρε, vε) of the diffuse-interface ∈ model, for a.e. τ [0, T ), including τ = 0, satisfies the energy equality 1 ∈ 1 τ2 tot tot Eε (τ2)+ Dε(t)dt = Eε (τ1) Zτ1 tot for all τ [τ , T ), where E is as in (3.2.9) and Dε is defined as 2 ∈ 1 ε 2 Dε = 2 µ(ρε) Dvε dx. | | ZΩ In view of Remark 3.2.6, we may apply Lemma 5.3.3. Then, for all τ W 1,1(0, T ) with ∈ τ 0 and τ(T ) = 0, we obtain ≥ T T tot tot ′ E (0)τ(0) + E (t)τ (t)dt Dε(t)τ(t)dt. (5.3.4) ε ε ≥ Z0 Z0 Next, we let ε 0 in (5.3.4): by Assumptions 5.1.1, we have → 2 tot,(i) tot n−1 (i) 1 (i) (i) lim Eε = lim Eε (0) = 2σst (Γ0 )+ 2 ρ0 v0 dx. ε→0 ε→0 H Ω Z

Using Lemma 5.2.1 and Corollary 5.3.2, we obtain T tot ′ lim Eε (t)τ (t)dt ε→0 Z0 T 1 ε 2 1 2 ′ = lim W (ρε)+ ρε + ρε vε dx τ (t)dt ε→0 ε 2 |∇ | 2 | | Z0 ZΩ T = 2σ n−1(Γ(t)) + 1 ρ v 2 dx τ ′(t)dt. stH 2 0 | 0| Z0 ZΩ To study the limiting behaviour of Dε(t) for ε 0, we use that, by Assumptions 3.2.1, µ → is non-negative and Lipschitz continuous, say with Lipschitz constant Lµ > 0. Therefore, there holds 4 2 2 µ(ρε) µ(ρ ) µ(ρε) µ(ρ ) µ(ρε)+ µ(ρ ) − 0 ≤ − 0 0 q q q q q q 2 = µ(ρε) µ(ρ ) | − 0 | 2 2 L ρε ρ . ≤ µ | − 0| ∞ 2 Since ρε ρ in L (0, T ; L (Ω)) as ε 0, it follows that → 0 → ∞ 4 µ(ρε) µ(ρ ) in L (0, T ; L (Ω)). → 0 q q 2 2 n×n Recalling from Theorem 3.4.1 that Dvε ⇀ Dv weakly in L (0, T ; L (Ω) ) for ε 0, 0 → we infer that

2 4 n×n µ(ρε)Dvε ⇀ µ(ρ0)Dv0 weakly in L (0, T ; L 3 (Ω) ) q q 2 2 n×n as ε 0. Additionally, ( µ(ρε)Dvε) is uniformly bounded in L (0, T ; L (Ω) ) by → ε∈(0,1) the energy equality (3.2.16)p and by Assumptions 3.3.1. We conclude that, after a possible passage to an appropriate subsequence, for ε 0, there holds → 2 2 n×n µ(ρε)Dvε ⇀ µ(ρ0)Dv0 weakly in L (0, T ; L (Ω) ). q q

123 Chapter 5 The Sharp-Interface Limit

Using the continuous embedding W 1,1(0, T ) ֒ L∞(0, T ) and τ 0 implies → ≥ 2 2 n×n µ(ρε)Dvε√τ ⇀ µ(ρ0)Dv0√τ weakly in L (0, T ; L (Ω) ). q q By the lower semi-continuity of the L2(0, T ; L2(Ω)n×n)-norm with respect to weak convergence, we obtain

T 2 T 2 lim inf µ(ρε)Dvε√τ dx dt µ(ρ0)Dv0√τ dx dt. ε→0 ≥ Z0 ZΩ Z0 ZΩ q q

Since τ is independent of x, this yields T T 2 2 lim inf µ(ρε(t)) Dvε(t) dxτ(t)dt µ(ρ0(t)) Dv0(t) dxτ(t)dt. ε→0 | | ≥ | | Z0 ZΩ Z0 ZΩ Hence, by letting ε 0 in (5.3.4), we obtain → 2 n−1 (i) 1 (i) (i) 2σst (Γ0 )+ 2 ρ0 v0 dx τ(0) H Ω Z T + 2σ n−1(Γ(t)) + 1 ρ (t) v (t) 2 dx τ ′(t)dt stH 2 0 | 0 | Z0 ZΩ T 2 µ(ρ (t)) Dv (t) 2 dxτ(t)dt. ≥ 0 | 0 | Z0 ZΩ Finally, in view of Remark 4.3.17, another application of Lemma 5.3.3 gives (5.3.3).

5.4 Regularity of Limiting Velocity and Transport Equation

Taking the weak formulation of the conservation-of-mass equation (3.2.14) to the limit ε 0 and using the energy estimate from Theorem 5.3.4, we may conclude that the → pair (ρ0, v0) has the desired regularity and that χ0 is a weak solution of the transport equation (4.3.24). To this end, we will use a technique due to DiPerna and Lions [36] and recall that distributional solutions of the conservation-of-mass equation are so-called renormalized solutions.

Lemma 5.4.1 (Renormalized solutions). Let s [1, ) and r [1, ] be such that ∈ ∞ ∈ ∞ 0 < 1 + 1 1. If a pair (ρ, v) Ls(Ω (0, T )) Lr(0, T ; W 1,r(Ω)n) with ρ 0 a.e. in s r ≤ ∈ × × ≥ Ω (0, T ) satisﬁes the transport equation ×

∂tρ + div(ρv) = 0 in Ω (0, T ) (5.4.1) × in the distributional sense, i.e., for every ϕ C∞(Ω (0, T )), there holds ∈ 0 × T ρ∂tϕ + ρv ϕ dx dt = 0, ·∇ Z0 ZΩ then (ρ, v) is a renormalized solution of (5.4.1) which means that, there holds

T ′ b(ρ)∂tϕ + b(ρ)v ϕ (ρb (ρ) b(ρ)) div(v)ϕ dx dt ·∇ − − Z0 ZΩ for any b C1([0, )) W 1,∞(0, ) and any ϕ C∞(Ω (0, T )). ∈ ∞ ∩ ∞ ∈ 0 ×

124 5.4 Regularity of Limiting Velocity and Transport Equation

Proof. See [40, Theorem 10.29].

Theorem 5.4.2 (Transport equation). Suppose that Assumptions 5.1.1 and 5.1.6 hold true.

2 1 n 1. Let v L (0, T ; H (Ω) ) be the limit function of the sequence (vε) as in The- 0 ∈ 0 ε∈(0,1) ∞ 2 orem 3.4.1. Then v0 belongs to L (0, T ; Lσ(Ω)).

∞ 2. The function χ0, given by (5.1.5), belongs to L (0, T ; BV (Ω)) and solves the transport equation

∂tχ + v χ = 0 in Ω (0, T ), 0 0 ·∇ 0 × (i) χ0(0) = χ0 in Ω (i) for prescribed initial data χ0 = χ −,(i) in the sense of Deﬁnition 4.3.13, i.e., for all Ω0 ϕ C∞ (Ω [0, T )), there holds ∈ (0) × T (i) χ (∂tϕ + v ϕ)dx dt + χ ϕ(0) dx = 0. (5.4.2) 0 0 ·∇ 0 Z0 ZΩ ZΩ Proof. The proof will be divided into several steps. Step 1. Firstly, we prove the stated regularity of the functions v and χ . Let t (0, T ). 0 0 ∈ In view of Corollary 3.5.6, there holds ρ (t) β > 0 a.e. in Ω. Then the energy estimate 0 ≥ 1 (5.3.3) yields

2σ n−1(Γ(t)) + 1 β v (t) 2 dx stH 2 1 | 0 | ZΩ 2σ n−1(Γ(t)) + 1 ρ (t) v (t) 2 dx ≤ stH 2 0 | 0 | ZΩ 2 n−1 (i) 1 (i) (i) 2σst (Γ0 )+ 2 ρ0 v0 dx. ≤ H Ω Z

The right-hand side of the latter inequality is bounded by Assumptions 5.1.1. Moreover, (2.3.4) yields χ (t) (Ω) = n−1(Γ(t)). This implies that χ L∞(0, T ; BV (Ω)) and |∇ 0 | H 0 ∈ v L∞(0, T ; L2(Ω)n). 0 ∈ ∞ Step 2. Let ϕ C (Ω [0, T )). As (ρε, vε) is a weak solution of (3.2.1)–(3.2.6), by ∈ (0) × (3.2.14), we have

T (i) ρε∂tϕ + ρεvε ϕ dx dt + ρ ϕ(0) dx = 0. ·∇ ε Z0 ZΩ ZΩ Letting ε 0, by Theorems 3.4.1 and 3.4.4 and Assumptions 5.1.1, it follows that → T (i) ρ ∂tϕ + ρ v ϕ dx dt + ρ ϕ(0) dx = 0. (5.4.3) 0 0 0 ·∇ 0 Z0 ZΩ ZΩ ∞ 2 Step 3. To prove that v0 belongs to L (0, T ; Lσ(Ω)), by Lemma 5.4.1, we conclude that for any b C1([0, )) W 1,∞(0, ) and any ϕ C∞(Ω (0, T )), there holds ∈ ∞ ∩ ∞ ∈ 0 × T ′ b(ρ )∂tϕ + b(ρ )v ϕ (ρ b (ρ ) b(ρ )) div(v )ϕ dx dt = 0. 0 0 0 ·∇ − 0 0 − 0 0 Z0 ZΩ

125 Chapter 5 The Sharp-Interface Limit

Choosing b such that b(β ) = b(β ) = 0, b′(β ) = 1 and b′(β ) = 1 and recalling that 1 2 1 β1 2 β2 ρ (t) β , β a.e. in Ω yields 0 ∈ { 1 2} T T v ϕ dx dt = div(v )ϕ dx dt = 0. (5.4.4) 0 ·∇ − 0 Z0 ZΩ Z0 ZΩ In particular, this means that div(v )=0inΩ (0, T ) in the distributional sense. By 0 × Lemma 4.3.2, we conclude that v L∞(0, T ; L2 (Ω)). 0 ∈ σ Step 4. It remains to verify that χ0 solves the transport equation. To this end, we use in (5.4.3) the representations ρ (t) = (β β )χ (t)+ β for t [0, T ] and 0 1 − 2 0 2 ∈ (i) (i) ρ0 = (β1 β2)χ −,(i) + β2 = (β1 β2)χ0 + β2 − Ω0 − from (5.1.1) and (5.1.4). This leads to

T (i) (β β ) χ (∂tϕ + v ϕ)dx dt + χ ϕ(0) dx 1 − 2 0 0 ·∇ 0 Z0 ZΩ ZΩ ! T = β ∂tϕ + v ϕ dx dt + ϕ(0) dx (5.4.5) − 2 0 ·∇ Z0 ZΩ ZΩ ! T = β v ϕ dx dt − 2 0 ·∇ Z0 ZΩ for every ϕ C∞ (Ω [0, T )). Due to (5.4.4), the right-hand side vanishes and we ﬁnally ∈ (0) × infer (5.4.2), which concludes the proof.

5.5 Variational Formulation

It remains to justify the claim that the pair (ρ0, v0) satisfies the variational formulation (4.3.25). To this end, as already announced in Section 3.6, we use in (3.2.15) divergence-free test functions ψ C∞([0, T ); C∞ (Ω)) and send ε 0. ∈ 0 0,σ → Theorem 5.5.1 (Variational formulation). If Assumptions 5.1.1, 5.1.6 and 5.3.1 are satisfied, then, for all ψ C∞([0, T ); C∞ (Ω)), there holds ∈ 0 0,σ T ρ v ∂tψ + ρ v v : ψ 2µ(ρ )Dv : Dψ dx dt 0 0 · 0 0 ⊗ 0 ∇ − 0 0 Z0 ZΩ T (5.5.1) = ρ(i)v(i) ψ(0) dx 2σ ν− ν− : ψ d n−1(x)dt. − 0 0 · − st ⊗ ∇ H ZΩ Z0 ZΓ(t) Proof. Let (ρε, vε) be a weak solution of (3.2.1)–(3.2.6) in the sense of Definition 3.2.5. Using in (3.2.15) divergence-free test functions ψ C∞([0, T ); C∞ (Ω)), we obtain ∈ 0 0,σ T (i) (i) 0= ρεvε∂tψ + ρεvε vε : ψ dx dt + ρ v ψ(0) dx ⊗ ∇ ε ε · Z0 ZΩ ZΩ T T 2µ(ρε)Dvε : Dψ dx dt + ε ρε ρε : ψ dx dt. − ∇ ⊗∇ ∇ Z0 ZΩ Z0 ZΩ As µ is Lipschitz continuous by Assumptions 3.2.1, µ(ρε) inherits the convergence proper-

ties of ρε. Hence, letting ε 0, we ﬁnally obtain (5.5.1) due to Assumptions 5.1.1 and → 5.3.1 and Theorems 3.4.1, 3.4.4 and 5.2.4.

126 5.6 Main Theorem

5.6 Main Theorem

Now we are able to formulate the main theorem of this thesis. We gather together the compactness properties of solutions to the diﬀuse-interface model obtained in Chapter 3 as well as the convergence results of the present chapter. As a consequence, we are able to investigate the sharp-interface limit for the phase-ﬁeld-like scaling of the Navier–Stokes– Korteweg equations, at the level of weak solutions along suitable subsequences.

Theorem 5.6.1 (Sharp-interface limit). Suppose that Assumptions 5.1.1, 5.1.6 and 5.3.1 are satisﬁed and let (ρεj , vεj )j∈N be a subsequence of (ρε, vε)ε∈(0,1). Then, there exist a N subsequence (ρεjm , vεjm )m∈ and a pair of limit functions (ρ0, v0) with

0, 1 2 ∞ ρ C 28 ([0, T ]; L (Ω)) L (0, T ; BV (Ω, β , β )) 0 ∈ ∩ { 1 2} and v L2(0, T ; H1(Ω)n) L∞(0, T ; L2 (Ω)) 0 ∈ 0 ∩ σ such that the following conditions are satisﬁed.

1. Convergence properties. Let t [0, T ]. For m , there holds ∈ →∞ 0, 1 2 a) ρε ρ in C 29 ([0, T ]; L (Ω)). jm → 0 q b) ρε (t) ρ (t) in L (Ω) for any q [1,p ). jm → 0 ∈ ∗ p∗ c) ρεjm (t) ⇀ ρ0(t) weakly in L (Ω).

2 1 n d) vεjm ⇀ v0 weakly in L (0, T ; H (Ω) ).

2. Solving the sharp-interface model. The pair (ρ0, v0) is a weak solution of the (i) (i) sharp-interface model (4.1.2)–(4.1.9) with respect to prescribed initial data (ρ0 , v0 ) in the sense of Deﬁnition 4.3.14.

Proof. The convergence properties of (ρε, vε)ε∈(0,1) have already been established in The- orems 3.4.1 and 3.4.4. The representation and the regularity of the limiting density ρ0 have been investigated in Corollary 3.5.6. In the present chapter we successively checked the conditions of Deﬁnition 4.3.14; see Theorems 5.3.4, 5.4.2 and 5.5.1. Therefore, (ρ0, v0) is a weak solution of the sharp-interface model (4.1.2)–(4.1.9).

5.7 Discussion: Convergence of Associated Energy Functional

Assuming the convergence of an associated energy functional to a suitable surface measure, we performed the sharp-interface limit for the phase-ﬁeld-like scaling of the Navier–Stokes– Korteweg model; see Theorem 5.6.1. This additional condition, see Assumptions 5.1.6, will

127 Chapter 5 The Sharp-Interface Limit

be discussed in this concluding section. For simplicity of presentation, we will consider the time-independent setting. Then Assumptions 5.1.6 read

1 ε 2 ∗ n−1 ∗ W (ρε)+ ρε ⇁ 2σ xΓ weakly- in C (Ω) , (5.7.1) ε 2 |∇ | stH ∗ 0 where Γ = ∂∗E for some set E Ω of ﬁnite perimeter. To simplify matters, we assume ⊂⊂ that β2 1 σst = 2 W (z)dz, (5.7.2) β Z 1 q which, for example, holds true for the canonical double-well potential introduced in Ex- ample 3.1.3, cf. also Remark 3.5.7. In many situations, it is convenient to replace (5.7.1) by the equivalent formulation

1 ε 2 n−1 lim W (ρε)+ ρε dx = 2σst (Γ). (5.7.3) ε→0 ε 2 |∇ | H ZΩ Remark 5.7.1. Assumptions in the fashion of (5.7.1) and (5.7.3) have for example been used by Le [65] and Luckhaus and Sturzenhecker [70].

In view of Theorem 5.6.1, condition (5.7.1) is sufficient to perform the sharp-interface limit. To start the discussion, in Section 5.7.1 we shall briefly recall how the latter assumptions contribute to the proof of Theorem 5.6.1. It is natural to ask, on the one hand, for situations where this condition can be recovered and proved rigorously and, on the other hand, to what extent this assumption can be weakened or even removed. In Section 5.7.2 we point out that (5.7.1) is satisfied in the static case (Section 5.7.2.1)

as well as in the case where we assume an asymptotic expansion of ρε in powers of ε (Section 5.7.2.2). In contrast, in Section 5.7.3 we will present some heuristic ideas as to why (5.7.1) may not hold true in general. We conclude the present section by brieﬂy discussing (5.7.1) in the context of varifolds in Section 5.7.4.

5.7.1 Why Do We Need Condition (5.7.1)?

Assuming condition (5.7.1) is essential for the analysis of the present chapter. We want to emphasize that in the proof of Theorem 5.6.1 we crucially used (5.7.1) in the two following respects.

5.7.1.1 Convergence of Capillary Term

According to the discussion in Section 3.6, an important point of our analysis is to control the convergence of the capillary term

ε ρε ρε : ψ dx (5.7.4) ∇ ⊗∇ ∇ ZΩ as ε 0. This term occurs in the weak formulation of the linear-momentum equation → (3.2.15), which merely incorporates divergence-free test functions ψ C∞ (Ω). Assump- ∈ 0,σ tion (5.7.1) allows us to control the limiting behaviour of (5.7.4) for ε 0 and to identify →

128 5.7 Discussion: Convergence of Associated Energy Functional

its limit. As proven in Theorem 5.2.10, due to (5.7.1), there holds

− − n−1 lim ε ρε ρε : ψ dx = 2σst ν ν : ψ d (x). (5.7.5) ε→0 ∇ ⊗∇ ∇ ⊗ ∇ H ZΩ ZΓ Recall from Lemma 4.3.8 that the right-hand side in (5.7.5) coincides with the mean- curvature functional 2σ κν− ψ d n−1(x), (5.7.6) st · H ZΓ in the case of smooth interfaces Γ. Therefore, (5.7.5) can be regarded as a weak formulation of (5.7.6). Identifying the weak-mean-curvature functional (up to multiplication of a constant) is crucial for obtaining a sharp-interface model incorporating the dynamic version of the Young–Laplace law (4.1.7).

5.7.1.2 Limit Passage in the Energy Inequality

The second regard in which (5.7.1) is essential is the passage to the limit ε 0 in the → energy inequality (3.2.16); see Theorem 5.3.4. In the proof of Lemma 5.2.1 we veriﬁed the lower asymptotic bound

n−1 1 ε 2 2σst (Γ) lim inf W (ρε)+ ρε dx. (5.7.7) H ≤ ε→0 ε 2 |∇ | ZΩ There, Assumption (5.7.1) was only used to ensure that the lower bound 2σ n−1(Γ) is stH “asymptotically sharp“, i.e., that there holds

1 ε 2 n−1 lim sup W (ρε)+ ρε dx = 2σ (Γ), (5.7.8) ε 2 |∇ | stH ε→0 ZΩ which is, due to the lower bound (5.7.7), equivalent to (5.7.3). This resolves the main difficulty in passing to the limit ε 0 in the energy inequality (3.2.16). Additionally, → (5.7.8) allows one to conclude that any cluster point (ρ0, v0) of (ρε, vε)ε∈(0,1) satisfies the energy inequality (4.3.26) which is an essential condition required by the weak formulation of the sharp-interface model; see Definition 4.3.14.

5.7.2 Situations, where Assumption (5.7.1) is Satisﬁed

We ﬁrstly explore the static case, see also Section 1.5, and secondly, we consider the method of formal asymptotic expansions.

5.7.2.1 Identiﬁcation of Limiting Measure in the Static Case

From an analytic point of view, it is convenient to consider the transformed functions

rε introduced in (3.3.4). In contrast to (ρε)ε∈(0,1), the sequence (rε)ε∈(0,1) is uniformly bounded in BV (Ω); see Lemma 3.3.6. By means of compactness arguments, after passing 1 to a suitable subsequence, which we do not relabel, as ε 0, there holds rε r in L (Ω) → → 0 and for the corresponding gradients

∗ n ∗ rε ⇁ r weakly- in (Ω) and rε ⇁µ weakly- in (Ω) ∇ ∇ 0 ∗ M |∇ | 0 ∗ M

129 Chapter 5 The Sharp-Interface Limit for some limiting measure µ (Ω). Naturally the question of the relation between the 0 ∈ M limiting measures r and µ arises. |∇ 0| 0 A positive answer is given by Luckhaus, Modica and Sternberg [69, 73, 92]. In the static case, they veriﬁed (5.7.3) for minimizers of the energy functional

1 ε 2 W (ρε)+ ρε dx ε 2 |∇ | ZΩ under the constraint of prescribed mass, by means of Γ-convergence techniques, cf. also Section 1.5. An important point of their analysis is to show that

µ = r = 2σ n−1xΓ. (5.7.9) 0 |∇ 0| stH Once the identiﬁcation (5.7.9) is established, one obtains (5.7.3) and with the help of Reshetnyak’s continuity theorem (Theorem 2.2.13), (5.7.5) also follows. Details of these arguments are carried out in the proof of Theorem 5.2.4. Summarising, (5.7.3) as well as the equivalent condition (5.7.1) are valid in the static case. The implications of (5.7.1) coincide for the static and the time-dependent case of the Navier–Stokes–Korteweg equations. However, for the time-dependent Navier–Stokes–

Korteweg equations we are neither able to identify the limiting measure µ0 nor to relate µ0 and r without any additional assumption and, hence, fail to prove (5.7.1) or (5.7.3). |∇ 0|

5.7.2.2 Optimal Proﬁle and Formal Asymptotic Analysis

A common technique for performing sharp-interface limits for phase-ﬁeld models is the method of formally matched asymptotic expansions. Among other assumptions, one sup- poses that there is a family of smooth evolving hypersurfaces Γ = (Γ(t))t∈[0,T ] separating − the domain Ω into two phases. As before, we denote by Ω (t) the phase related to β1 + and by Ω (t) the phase related to β2. The time-evolution of the interface Γ(t) can be described in terms of the signed distance d ( ,t) to Γ(t) by Γ · Γ(t)= x Ω:d (x,t) = 0 , ∈ Γ where the sign of d ( ,t) is chosen such that d ( ,t) > 0 if x Ω+(t). Furthermore, it Γ · Γ · ∈ is assumed that the quantities ρε and vε can be approximated by truncations of formal expansions in formal power series in ε, say ∞ m (m) ρε( ,t)= ε ρ ( ,t) (5.7.10) · m · X=0 away from the interface in Ω Γ δ (t), where Γδ(t)= x Ω : dΓ(x,t) δ is the tubular \ 2 ∈ | |≤ neighbourhood of Γ(t) of thickness δ > 0. The functions ρ(m), m N , are supposed ∈ to have real analytic extensions on Ω−(t) and on Ω+(t), but, in general, they are non- smooth across the interface Γ(t). In the interfacial layer Γδ(t) we may introduce new local coordinates by x = s + εzν(s,t),

130 5.7 Discussion: Convergence of Associated Energy Functional where s = s(x,t) is the projection of x on Γ(t) along the outer normal ν( ,t) of Ω−(t) · pointing into Ω+(t) and 1 z = z(x,t)= d (x,t). (5.7.11) ε Γ Furthermore, the existence of another formal power series ∞ m (m) ρε(x,t) =ρ ˆε(s,z,t)= ε R (s,z,t) (5.7.12) mX=0 that is valid near the interface in the interfacial region Γδ in the new coordinates (s,z,t) is assumed. The choice (5.7.11) reﬂects the fact that the thickness of the interfacial layer is proportional to ε for the inner expansions to be valid. The expansions (5.7.10) and

(5.7.12) are, respectively, referred to as the outer and the inner expansion of ρε. In view of Theorem 5.6.1, it is natural to assume that in the outer expansion we can identify the leading order term ρ(0) in (5.7.10) as

β if x Ω−(t), (0) 1 ρ (x,t)=  ∈ β if x Ω+(t).  2 ∈ Supposing that there is an overlapping region, where both expansions are valid, the two expansions coincide and therefore can be formally equated, i.e., matched order by order. This leads to the so-called matching conditions as one comes out of the inner region (z ) and into the inner region (x ξν, ξ 0). In particular, there holds → ±∞ ± ց (0) (0) lim R (s,z,t) = lim ρ (x ξν,t)= β1 z→−∞ ξց0 − and, analogously, (0) (0) lim R (s, z) = lim ρ (x + ξν,t)= β2, z→∞ ξց0 cf. for example [41, page 3, equation (5a)]. For details of this matching procedure we refer to [41, Chapter 1, Section 1] or [5, Section 4]. For this discussion, as above, we skip the time-dependence of functions. Additionally, we suppose that we can identify the leading order term R(0) in the inner expansion (5.7.12) as

(0) 1 R (s, z)=Θ0(z)=Θ0( ε dΓ(x)), where Θ0 is the so-called optimal proﬁle as in the following lemma.

Lemma 5.7.2 (Optimal proﬁle). The diﬀerential equation

Θ′′ + W ′(Θ ) = 0 in R, − 0 0 1 Θ0(0) = 2 (β1 + β2), (5.7.13) lim Θ0(z)= β1, z→−∞

lim Θ0(z)= β2 z→∞ admits a unique solution Θ C2(R). Moreover, Θ has the subsequent properties. 0 ∈ 0

131 Chapter 5 The Sharp-Interface Limit

1. There are constants C,τ > 0 such that 0 < Θ′ (z) C exp( τ z ) for any z R. 0 ≤ − | | ∈ 2. For every ϕ C1(Ω), there holds ∈ 1 ′ 1 2 n−1 lim Θ0( dΓ) ϕ dx = 2σst ϕ d (x), (5.7.14) ε→0 ε ε H ZΩ ZΓ where 1 ∞ σ = Θ′ (z)2 dz. (5.7.15) st 2 0 Z−∞ Proof. See [88, Lemma 2.6.1 and equation (5.22)].

Remark 5.7.3 (Capillary constant). Note that, in view of (5.7.13), the definitions of the constant σst in (5.7.2) and (5.7.15) coincide. This can be seen as follows. From the differential equation (5.7.13), we infer that, for z R, there holds ∈ 1 Θ′ (z)2 ′ =Θ′′(z)Θ′ (z)= W ′(Θ (z))Θ′ (z)= W (Θ (z)) ′. 2 0 0 0 0 0 0 Thus there is a constant c R such that 1 Θ′ (z)2 = W (Θ (z)) + c for all z R. By ∈ 2 0 0 ∈ (5.7.13) and the convergence properties from Lemma 5.7.2, we conclude that c = 0. This, in turn, means that 1 Θ′ (z)2 = W (Θ (z)) (5.7.16) 2 0 0 for any z R. Taking into account Lemma 5.7.2 gives ∈ ∞ ∞ β2 1 ′ 2 ′ 1 σst = Θ0(z) dz = Θ0(z) W (Θ0(z)) dz = 2 W (z)dz. 2 −∞ −∞ β Z Z q Z 1 q Consequently, the two definitions of σst are compatible.

We assume that for ρε a composite expansion

1 ρε(x)=Θ ( d (x)) + ε(x) for x Ω, (5.7.17) 0 ε Γ R ∈

is valid such that the remainder term ε is uniformly bounded on Ω and for ε 0 R → asymptotically vanishes in the following sense

1 2 ε(x) dx 0 and ε ε(x) dx 0. (5.7.18) ε |R | → |∇R | → ZΩ ZΩ It is worth pointing out that a composite expansion of the form (5.7.17) is typically obtained in the framework of formal asymptotic expansions for phase-ﬁeld models. From

the special form (5.7.17)–(5.7.18) of ρε, we can recover (5.7.3) and thus (5.7.1) is satisﬁed. This implies the crucial result of Lemma 5.2.1, including the equipartition of energy (5.2.2). Furthermore, we conclude the convergence of the capillary term to the weak curvature functional (5.7.5), i.e., the result of Theorem 5.2.4. Let us formulate this precisely in the following theorem.

Theorem 5.7.4. Let (5.7.17) and (5.7.18) be valid. Then the following convergence properties are satisﬁed.

132 5.7 Discussion: Convergence of Associated Energy Functional

1. Convergence of total mass. Let σst be given by (5.7.2) and let Γ be as in (5.7.1). Then there holds

1 ε 2 n−1 lim W (ρε)+ ρε dx = 2σst (Γ). (5.7.19) ε→0 ε 2 |∇ | H ZΩ 2. Equipartition of energy.

ε 2 1 lim 2 ρε ε W (ρε) dx = 0. (5.7.20) ε→0 Ω |∇ | − Z

3. Convergence of capillary term. For every ψ C ∞(Ω)n, there holds ∈ 0 n−1 lim ε ρε ρε : ψ dx = 2σst ν ν : ψ d (x). ε→0 ∇ ⊗∇ ∇ ⊗ ∇ H ZΩ ZΓ Proof. Let ε (0, 1). ∈

1. From the special form (5.7.17) of ρε, we deduce

2 1 ′ 1 2 2 ′ 1 2 ε ρε dx = Θ ( d ) d + 2Θ ( d ) d ε + ε ε dx. |∇ | ε 0 ε Γ |∇ Γ| 0 ε Γ ∇ Γ · ∇R |∇R | ZΩ ZΩ Using that d = 1 on Γ, cf. [20, Theorem 1.5], in view of Lemma 5.7.2 and (5.7.18), |∇ Γ| we infer that

ε 2 2 n−1 n−1 lim ρε dx = σst dΓ d (x)= σst (Γ). (5.7.21) ε→0 2 |∇ | |∇ | H H ZΩ ZΓ ∗ ∗ Let x Ω and let > 0 be a constant, independent of x and ε, such that ε(x) . ∈ R |R | ≤ R 1 ∗ 1 ∗ Then there exists some ξε = ξε(x) [Θ ( d (x)) , Θ ( d (x)) + ] such that ∈ 0 ε Γ − R 0 ε Γ R 1 1 ′ W (Θ ( d (x)) + ε(x)) W (Θ ( d (x))) = W (ξε(x)) ε(x) 0 ε Γ R − 0 ε Γ R by the mean-value theorem. Then we conclude that

1 ′ W (ρε(x)) W (Θ ( d (x))) = W (ξε(x)) ε(x) − 0 ε Γ R ′ sup W (z) ε(x) , ∗ ∗ ≤ z∈[β1−R ,β2+R ] |R |

since β Θ β due to Lemma 5.7.2. By (5.7.18), we obtain 1 ≤ 0 ≤ 2 1 1 lim W (ρε) W (Θ0( ε dΓ)) dx = 0. ε→0 ε Ω − Z

Taking into account (5.7.16) and Lemma 5.7.2, this implies

1 1 1 lim W (ρε)dx = lim W (Θ0( dΓ)) dx ε→0 ε ε→0 ε ε ZΩ ZΩ 1 ′ 1 2 (5.7.22) = lim Θ0( dΓ) dx ε→0 2ε ε ZΩ = σ n−1(Γ). stH Finally, addition of (5.7.21) and (5.7.22) gives (5.7.19).

133 Chapter 5 The Sharp-Interface Limit

2. As in the proof of Lemma 5.2.1, by (5.7.19), we conclude (5.7.20).

3. Let ψ C∞(Ω)n. Due to (5.7.17), we have ∈ 0

ε ρε ρε : ψ dx ∇ ⊗∇ ∇ ZΩ 1 ′ 1 2 ′ 1 = Θ ( d ) d d + 2Θ ( d ) d ε + ε ε ε : ψ dx. ε 0 ε Γ ∇ Γ ⊗∇ Γ 0 ε Γ ∇ Γ ⊗ ∇R ∇R ⊗ ∇R ∇ ZΩ Taking into account (5.7.18) as well as Lemma 5.7.2 and the fact that d = ν on Γ, ∇ Γ cf. [20, Theorem 1.5], we conclude that

′ 1 lim Θ0( dΓ) dΓ ε : ψ dx = lim ε ε ε : ψ dx = 0 ε→0 ε ∇ ⊗ ∇R ∇ ε→0 ∇R ⊗ ∇R ∇ ZΩ ZΩ and ﬁnally, by (5.7.14), we infer that

n−1 lim ε ρε ρε : ψ dx = 2σst dΓ dΓ : ψ d (x) ε→0 ∇ ⊗∇ ∇ ∇ ⊗∇ ∇ H ZΩ ZΓ = 2σ ν ν : ψ d n−1(x). st ⊗ ∇ H ZΓ This concludes the proof.

5.7.3 Heuristic Picture

In Section 5.7.2 it was possible to justify the convergence assumption (5.7.1) in special situations. Here, we shall give some heuristic arguments as to why this additional assumption, in general, may not hold true and therefore cannot be disregarded. The subsequent discussion incorporates the heuristic ideas of Abels [2], who studied the special case of the sharp-interface model (4.1.2)–(4.1.9) with equal densities β1 = β2 = 1. To simplify matters, we consider the level sets

1 Γε = x Ω : ρε(x)= (β + β ) ∈ 2 1 2 rather than diﬀuse interfaces, e.g., tubular neighbourhoods of Γε of order ε. Both kinds of interfaces share the same qualitative behaviour in the limit ε 0. We assume that, as → ∗ ε tends to 0, Γε converges in an appropriate sense to a limiting interface Γ = ∂ E, where E Ω is a set of ﬁnite perimeter in Ω. Clearly, we expect the set E to satisfy ⊂

ρ = (β β )χE + β a.e. in Ω. 0 1 − 2 2 Due to the following oscillation and concentration eﬀects at Γ, we cannot hope for the identiﬁcation (5.7.9) in general.

5.7.3.1 Hidden Interfaces

Several parts of the level sets Γε might meet in the limit ε 0. This phenomenon, referred → to as “hidden” or “phantom” interfaces, is sketched in Figure 5.1. An illustrative example are two circles (n = 2) or two bubbles (n = 3) with positive distance for ε (0, 1) touching ∈ each other in the limit ε 0. →

134 5.7 Discussion: Convergence of Associated Energy Functional

Ω−(t)Ω−(t) Ω−(t) Ω−(t) ε 0 →

Figure 5.1: Hidden interface [2].

5.7.3.2 Oscillations

The level sets Γε might incorporate some oscillation eﬀects. In the limit ε 0 these → oscillations might vanish and, as a consequence, the interface area might be reduced, cf. Figure 5.2.

Ω+(t) Ω+(t)

ε ≈ ε 0 ε → ≈ Ω−(t) Ω−(t)

Figure 5.2: Oscillation eﬀects [2].

5.7.3.3 Emulsiﬁcation

The level sets Γε might form spikes with distance of order ε as illustrated in Figure 5.3. As ε tends to zero, the distance of nearby spikes reduces and in the limit this leads to concentration eﬀects similar to emulsiﬁcation. ε ≈

ε 0 → ε ≈

Figure 5.3: Concentration eﬀects [2].

135 Chapter 5 The Sharp-Interface Limit

5.7.4 Varifold Approach

The mathematical treatment of the eﬀects described in the previous Section 5.7.3 is an open problem. We want to stress that (5.7.1) excludes these kinds of phenomena and thus allows one to take the sharp-interface limit to a physically meaningful sharp-interface model.

5.7.4.1 Multiplicity of Interfaces

In general, we expect the limiting measure in (5.7.1) to have the form

2σ m(x) n−1xΓ, st H where m(x) 1, x Γ, is called the multiplicity of the interface Γ. The choice of the ≥ ∈ limit measure 2σ n−1xΓ in (5.7.1) corresponds to the situation where Γ is an interface stH of constant multiplicity m 1, cf. [65]. In particular, it excludes the occurrence of hidden ≡ interfaces. See [29, 65] for a more detailed discussion of this issue. The multiplicity of the limiting interface reﬂects how many times the set

1 x Ω : ρε(x)= (β + β ) { ∈ 2 1 2 } folds into the limiting interface Γ. The question of (identifying) the multiplicity of the interface is related to the control of the chemical potential

1 ′ wε = ε∆ρε + W (ρε). − ε Tonegawa showed, for the time-independent case, that the limiting interface has constant 1,r n multiplicity one if the chemical potential is uniformly bounded in W (Ω) for r > 2 [94, 95]. For example, in the case of the Cahn–Hilliard equation, due to appropriate a priori estimates on the chemical potential, those bounds are available [29]. By contrast, for the Allen–Cahn equation, the control over the chemical potential is too weak: the multiplicity of the limiting interface can be any positive integer [28]. Another counterexample is given by Abels and Lengeler [6]. In the case of the Navier–Stokes–Korteweg equations, treated here, we could neither derive a priori estimates allowing for control of the chemical potential nor give a counterexample.

5.7.4.2 Generalised Varifold Solutions

The correct treatment of interfaces with general multiplicity m(x) and of possible oscillation or concentration effects, as heuristically described in Section 5.7.3, is an open question. One possible way to handle these phenomena mathematically might be to work in a varifold setting. Plotnikov [78] was first to use the notion of varifold to construct generalised solutions of a classical sharp-interface model for two-phase flow. This approach was extended by

136 5.7 Discussion: Convergence of Associated Energy Functional

Abels [1]. Ilmanen [55, 29] and Chen [55, 29] studied the sharp-interface limit for the Allen– Cahn and the Cahn–Hilliard equation, respectively, in a varifold context. Subsequently, these techniques have been applied to a coupled Navier–Stokes/Cahn–Hilliard system due to Hohenberg and Halperin [53] (also known as “model H”) by Abels and R¨oger [7] and to a generalisation of model H by Abels and Lengeler [6]. In the last two works it is crucial that energy inequalities lead to a priori estimates of the gradient of the chemical potential in L∞(0, ; L2(Ω)n) and L2(0, ; L2(Ω)n), respectively. ∞ ∞

137

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