Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations Dissertation zur Erlangung des Doktorgrades vorgelegt von Johannes Daube an der Fakult¨at f¨ur Mathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg Februar 2017 Dekan: Prof. Dr. Gregor Herten 1. Gutachter: Prof. Dr. Dietmar Kr¨oner 2. Gutachter: Prof. Dr. Helmut Abels Datum der m¨undlichen Pr¨ufung: 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 thick- ness 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]. v 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 Kr¨oner, 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 N¨agele 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.