Function Spaces, Time Derivatives and Compactness for Evolving Families of Banach Spaces with Applications to Pdes

Home , Bochner space, Weak derivative

arXiv:2105.07908v1 [math.AP] 17 May 2021 ieeovn ohe spaces Bochner Time-evolving 3 functions space-valued Banach on Background 2 Theory I h ui–in em neovn spaces evolving in lemma Aubin–Lions The 6 triple Gelfand The 5 spaces evolving in derivatives Time 4 Introduction 1 Contents ∗ † § ‡ . rtrafreovn pc qiaec ...... space . . . . the . . for . . . . equivalence . . . . space . . Evolving . . . ( . . for . . . . criteria 5.4 . equivalence . . Alternative . . space . . evolving theorem . . . for . . transport 5.3 . . . Criteria . product: . . . . inner . . . the . . . 5.2 . Differentiating ...... 5.1 ...... equivalence . . . . . space . . . . . evolving . theorem . . for . . transport . . . Criteria . . . product: . . . . . duality . . . . . the . 4.6 . . Differentiating ...... derivative spaces . . . time . Sobolev–Bochner . 4.5 . . weak . Evolving . remarks . the . . . . further of . . . and . . characterisation . 4.4 . functions . A . derivative . . smooth time . . . for . weak . . formula . the 4.3 . . Transport . of . . . . . properties . . . and . . 4.2 . Definition ...... 4.1 . . . space . . . Hilbert . . of . . . fields . . . measurable . . to . . . Relation ...... spaces . 3.2 . . Dual ...... 3.1 ...... triples Gelfand . product duality . the 2.3 Differentiating derivative time weak 2.2 The 2.1 ahmtc nttt,Uiest fWrik oetyCV Coventry Warwick, of University Institute, Mathematics eesrs nttt,Mhesrse3,117Bri,Ge Berlin, 10117 39, Mohrenstrasse Institute, Weierstrass ahmtc nttt,Uiest fWrik oetyCV Coventry Warwick, of University Institute, Mathematics ntttfu ahmtk ri nvri¨tBri,Arni Universität Berlin, Freie für Mathematik, Institut ucinsae,tm eiaie n opcns o evol for compactness and derivatives time spaces, Function fwl oens o ls fnniermntn problems monotone nonlinear of evolutionary class the a particular for space posedness well Sobolev–Bochner of aforementioned functi defini the the of with provide explicitly examples properties we concrete which Aubin–L for several hypersurfaces An analyse and then spaces. We Sobolev–Bochner func proved. classical differentiable the weakly of to spaces inne relate an the of space which availability Banach under the on ditions of rely families not does evolving which on derivative posed problems ational vltoaypolm htcnb prpitl formulate appropriately be can that problems evolutionary mlAlphonse Amal edvlpafntoa rmwr utbefrtetreatmen the for suitable framework functional a develop We aiiso aahsae ihapiain oPDEs to applications with spaces Banach of families X ∗ ( t ) p ⊂ Lpaeeuto namvn oano ufc)adidentif and surface) or domain moving a on equation -Laplace ig Caetano Diogo 5.8 H ( ...... ) t ) ⊂ L X p X ∗ ( t ) a 8 2021 18, May setting † W ale9 49 eln emn ( Germany Berlin, 14195 9, mallee Abstract ( mn ( rmany ,H X, 1 A,Uie igo ( Kingdom United 7AL, 4 A,Uie igo ( Kingdom United 7AL, 4 n Djurdjevac Ana ...... ) ihteasrc etn eeoe nti work. this in developed setting abstract the with d [email protected] rdc rHleta tutr n xlr con- explore and structure Hilbertian or product r ino h iedrvtv n eiyisomorphism verify and derivative time the of tion .W ocuewt h omlto n proof and formulation the with conclude We s. in wt ausi neovn aahspace) Banach evolving an in values (with tions oscmatesrsl nti etn salso is setting this in result compactness ions .W rps ento o h ektime weak the for definition a propose We s. na btateovn pc gnrlsn in (generalising space evolving abstract an on nsae vrtm-vligsaildomains spatial time-evolving over spaces on fprildffrnileutosadvari- and equations differential partial of t ...... ‡ ...... [email protected] [email protected] [email protected] ...... hre .Elliott M. Charles ) ...... oeadditional some y ...... ving ) § ) 25 22 10 25 24 23 23 17 16 15 14 12 10 ) 4 7 4 2 9 8 6 5 4 II Applications 26

7 Examples of function spaces on evolving domains and surfaces 26 7.1 L2(M(t))pivotspace ...... 28 7.2 H1(M(t))pivotspace ...... 29 7.3 H−1(Ω(t))pivotspace...... 30 7.4 Non-Gelfand triple examples ...... 34 7.5 Proofs of compatibility and evolving space equivalence ...... 36 7.5.1 L2 pivotspace ...... 36 7.5.2 H1 pivotspace ...... 37 7.5.3 H−1 pivotspace ...... 39

8 Well-posedness for a nonlinear monotone equation 42 8.1 Proofofwell-posedness...... 43

A Technical results 46

1 Introduction

In this paper, we provide a theory and analysis of time-dependent function spaces suitable for posing and solving evolutionary variational problems on families of time-evolving Banach spaces. We further demonstrate our theory via examples and applications of partial diﬀerential equations on moving domains and surfaces. By way of illustration, for each t ≥ 0, let H(t) be a Hilbert space and X(t) be a Banach space with dual X∗(t) such that X(t) ⊂ H(t) ⊂ X∗(t) is a Gelfand triple. We say that H(t) is the pivot space. Let A(t): X(t) → X∗(t) be an elliptic operator andu ˙ an appropriate time derivative (to be deﬁned later) of u. With this, we can consider the abstract problem

u˙(t)+ A(t)u(t)= f(t) in X∗(t), (1.1) u(0) = u0 in H(0).

One possible weak formulation concept for this problem would ask for the solution to satisfy

T T hu˙(t), η(t)iX∗ (t),X(t) + hA(t)u(t), η(t)iX∗ (t),X(t) = hf(t), η(t)iX∗ (t),X(t) Z0 Z0 for every appropriate test function η, as well as the satisfaction of the initial condition. To make this precise, one needs to specify (i) the exact functional spaces that the solutions lie in, (ii) how to define the time derivative in an abstract evolving Banach space setting, (iii) the properties of the above-mentioned spaces and objects that allow for analysis (e.g. existence of solutions) to be performed. Our motivation comes from the study of partial differential equations on moving or evolving domains and manifolds. Such equations have received considerable attention in part due to their wide applicability in the biological and physical sciences. We mention applications in biomembranes [55], cell interactions [5], cardio- vascular biomechanics [42], fluid mechanics [14], chemotaxis [30], to name but a few. In addition to modelling aspects, the analysis [1–3, 5, 17, 18, 22, 33, 35, 36] and numerics and simulation [27–29, 32, 43, 45, 46, 57] of such problems is challenging and an active area of research. In the case that X(t) is a Hilbert space, such issues have been considered. In particular, in [3] an abstract framework for the formulation and well posedness of solutions of equations of the form (1.1) was provided for linear parabolic problems in the Hilbert triple setting; for this, Lions-type solution spaces Wp,q(X,X∗) (referring to the set of p-integrable functions that have values in X(t) with q-integrable weak time derivatives with values in X∗(t)) were defined and rigorously justified to have certain properties that are necessary for the existence theory. See also [4] for several concrete examples of applications of this theory. In this work, our setup involves not only Gelfand triples but also more general families of Banach spaces

X(t) ⊂ Y (t) with no intermediate inner product structure available. As there is no pivot space to work with, the formulation and properties of the weak time derivative and evolving function spaces become more complicated. It is the

2 aim of this paper to provide the theoretical background for constructing these spaces in the fully Banach space setting, to study their properties, and to provide examples that will cover most cases of interest to practitioners working with evolutionary variational problems on moving domains and surfaces. We will also provide an Aubin– Lions type compactness result (a widely used tool in the study of nonlinear problems) for these spaces. A crucial point in achieving the Aubin–Lions result (as well as other results and properties) is an intermediary result in which we give conditions under which the space Wp,q(X, Y ) is isomorphic to the standard Sobolev–Bochner space (or Lions space) p,q p ′ q W (X0, Y0) := {u ∈ L (0,T ; X0): u ∈ L (0,T ; Y0)}, where X0 := X(0) and Y0 := Y (0). Expending effort in achieving this isomorphism property is worthwhile since p,q it has the advantage of allowing for a simple transferral of the properties of W (X0, Y0) onto the time-evolving version Wp,q(X, Y ). In particular, it leads to a relatively straightforward proof for the extension of the standard Aubin–Lions result to the evolving setting. In summary, the novelty of the work is the following: (1) we consider and define weak time derivatives in a fully Banach space setting (separability and reflexivity are not assumed); with no inner product or Gelfand triple structure to aid us, the formulation of such a time derivative is non-trivial and requires care and justification; (2) we provide conditions that can be checked to ensure the isomorphism with equivalence of norms between p,q p,q the standard Sobolev–Bochner space W (X0, Y0) and the evolving Sobolev–Bochner spaces W (X, Y ) under consideration in this paper; (3) we provide an Aubin–Lions result also in this generality (with no restriction needed for the evolving spaces to be related to domains or manifolds); (4) we study a number of concrete examples involving function spaces of moving domains and surfaces that fit our abstract framework; (5) we prove existence and uniqueness of solutions to a monotone first-order evolution equation (of the form (1.1)) in a Gelfand triple setting using the theory developed in this paper. Under the assumptions on the evolution of the spaces in this paper, it is always possible to pull back equations such as (1.1) onto a reference space X0 and apply standard theory on fixed spaces once the relevant assumptions have been verified. However, our approach — which enables the problem to be treated directly in its natural formulation — offers a certain elegance and simplicity and is also of use in numerical and finite element analysis [29] on moving domains/surfaces (in addition to being an interesting mathematical problem in its own right). Furthermore, pulling back onto a reference domain nonetheless requires the checking of regularity of the resulting coefficients in order to apply standard theory and the analogue of this is performed for some rather general cases in §7, which we believe has a wide appeal for a variety of problems on moving domains and surfaces.

Organisation of the paper. The paper is split into two parts. Part I focuses on the abstract theory and Part II contains applications of the theory. Beginning in §2 we recall some standard results about Bochner spaces and weak time derivatives and we provide a formula for differentiating the duality product of weakly p differentiable functions. In §3, we define and study properties of the evolving Bochner spaces LX and their dual spaces. We move onto defining a weak time derivative in §4, as well as defining spaces of functions with weak time derivatives and their relation to the standard Sobolev–Bochner spaces. We study the conditions under which the two spaces are isomorphic. Proceeding in §5, we specialise the above theory to the setting where we have a Gelfand triple, which leads to a simplification in the statement of the assumptions that are required. We generalise the Aubin–Lions result to our setting in §6, concluding Part I. In §7, we study several concrete examples of the abstract theory. Finally, in §8, we provide an application to a nonlinear parabolic equation.

Notation and conventions

• We will always work with real Banach spaces. • The action of the linear map x∗ ∈ X∗ on x ∈ X is denoted by

∗ ∗ hx , xiX∗,X = hx, x iX,X∗ . d c .Continuous, dense and compact embeddings of spaces will be denoted by ֒→, ֒−→ and ֒−→ respectively • T T • We will usually leave out the diﬀerential in integrals, i.e., we write 0 f(t) rather than 0 f(t) dt. • For a function f : [0,T ] → X onto a Banach space, we denote the diﬀerenceR quotient R f(t + h) − f(t) δ f(t) := . h h

3 • Given a,b ∈ R, a ∧ b := min(a,b). • The letters p and q will typically be used for (not necessarily conjugate) integrability exponents in Lp-type spaces; the conjugate of p will always be denoted by p′ := p/(p − 1).

Part I: Theory

2 Background on Banach space-valued functions

It is convenient to recall a few spaces of (vector-valued) functions and related concepts; useful texts on this topic ∞ are [8, 10, 19, 48, 59]. Firstly, we write D(Ω) ≡ Cc (Ω) to refer to the set of inﬁnitely diﬀerentiable functions Rn ∞ with compact support in the open set Ω ⊂ . Likewise, for a Banach space X, D((0,T ); X) ≡ Cc ((0,T ); X) denotes the space of smooth, compactly supported functions on (0,T ) with values in X. The dual space of D(Ω) will be denoted D∗(Ω), which is the space of continuous linear functionals on D(Ω) (i.e., the space of distributions) endowed with the strong dual topology. The space D∗((0,T ); X) will stand for the space of continuous linear mappings from D(0,T ) into X, i.e.,

D∗((0,T ); X)= L(D(0,T ),X), see [8] for further details. We will assume familiarity with the standard Bochner spaces Lp(0,T ; X).

2.1 The weak time derivative Given Banach spaces X ⊂ Y , a function u ∈ L1(0,T ; X) has a weak time derivative u′ ∈ L1(0,T ; Y ) if

T T u(t)ϕ′(t)= − u′(t)ϕ(t) ∀ϕ ∈ D(0,T ). (2.1) Z0 Z0 The integrals here are Bochner integrals and (2.1) is an equality in Y . See for example [59, §23.5] and [10, Deﬁnition II.5.7]. Remark 2.1. It is not necessary for the embedding X ⊂ Y to be via the inclusion map X ∋ x 7→ x ∈ Y . Indeed, (supposing that ι: X ֒→ Y is a continuous linear embedding, the deﬁnition of the weak derivative as in (2.1 should be interpreted in the following sense: a function u ∈ L1(0,T ; X) has a weak derivative u′ ∈ L1(0,T ; Y ) if

T T ι u(t)ϕ′(t) dt = − u′(t)ϕ(t) dt ∀ϕ ∈ D(0,T ). Z0 ! Z0 It is easy to see that the classical derivative of a smooth function u ∈ C1([0,T ]; X) is also a weak time derivative, and that weak time derivatives are unique. The former is a consequence of the Leibniz rule and the latter follows from the next result. Lemma 2.2. Let u ∈ L1(0,T ; X) be such that

T ∗ hu(t), ϕ(t)iX,X∗ =0 ∀ϕ ∈ D((0,T ); X ). Z0 Then u(t)=0 for almost all t ∈ [0,T ]. Proof. Take ϕ(t) = ψ(t)w with w ∈ X∗ and ψ ∈ D(0,T ). Then, from the fundamental lemma of the calculus of variations, we have for almost all t ∈ [0,T ],

hu(t), wiX, X∗ =0. Since w is also arbitrary in X∗, a corollary to the Hahn–Banach theorem now implies u(t) = 0 for such t. Remark 2.3. Brieﬂy, let us relate the weak time derivative to vector-valued distributional derivatives. For any 1 ∗ function u ∈ Lloc(0,T ; X) we can deﬁne a distribution Tu ∈ D ((0,T ); X) [19, Prop. XVIII.4] by

T ∗ Tu(ϕ) := u(t)ϕ(t), ∀ϕ ∈ D((0,T ); X ). Z0

4 By [19, Def. XVIII.3], the derivative of any distribution T ∈ D∗((0,T ); X) is a distribution T ′ ∈ D∗((0,T ); X) deﬁned by the formula T ′(ϕ) := −hT, ϕ′i ∀ϕ ∈ D(0,T ). Hence, the weak derivative of a function u ∈ L1(0,T ; X) is the unique element u′ ∈ L1(0,T ; Y ) such that ′ ′ Tu = Tu (the equality is between vector-valued distributions). Now let X ֒→ Y be a continuous embedding. We will be particularly interested in the space of functions u ∈ Lp(0,T ; X) with weak time derivative u′ ∈ Lq(0,T ; Y ) where p, q ∈ [1, ∞] are not necessarily conjugate exponents. Deﬁne the Sobolev–Bochner space

Wp,q(X, Y )= {u ∈ Lp(0,T ; X): u′ ∈ Lq(0,T ; Y )} (2.2) with the norm

′ kukWp,q(X,Y ) := kukLp(0,T ;X) + ku kLq(0,T ;Y ).

From [48, Lemma 7.1], (Wp,q(X, Y ) ֒→ C0([0,T ]; Y ), (2.3 and the following approximation result, the proof of which can be found in [48, Lemma 7.2], will be useful throughout the text: d (C1([0,T ]; X) ֒−→Wp,q(X, Y ). (2.4

2.2 Diﬀerentiating the duality product We now aim to obtain a general transport theorem (in the non-moving setting) in the spirit of a result in [53, Lemma 1.1, §III] characterising the weak time derivative by considering the derivative of an associated duality product. We start with the smooth version of the result. Lemma 2.4. Let X be a Banach space. Given u ∈ C1([0,T ]; X) and v ∈ C1([0,T ]; X∗), the map

t 7→ hu(t), v(t)iX,X∗ is absolutely continuous and

d ′ ′ hu(t), v(t)i ∗ = hu (t), v(t)i ∗ + hu(t), v (t)i ∗ for almost all t ∈ [0,T ]. dt X,X X,X X,X Proof. We begin with

1 u(t + h) − u(t) v(t + h) − v(t) (hu(t + h), v(t + h)iX,X∗ − hu(t), v(t)iX,X∗ )= , v(t + h) + u(t), . h h ∗ h ∗ X,X X,X Now we use the fact that by deﬁnition, the diﬀerence quotient

u(t + h) − u(t) δ u(t) := → u′(t) in X as h → 0 h h and v(t + h) → v(t) in X∗ to get

1 ′ ′ (hu(t + h), v(t + h)i ∗ − hu(t), v(t)i ∗ ) → hu (t), v(t)i ∗ + hu(t), v (t)i ∗ h X,X X,X X,X X,X which proves the desired formula. Since the derivative exists everywhere and is integrable, it follows that the duality product being diﬀerentiated is absolutely continuous. Arguing by approximation we obtain the following general transport theorem. ∋ Proposition 2.5. Let p, q ∈ [1, ∞] and X ֒→ Y be a continuous embedding of Banach spaces. For u ′ ′ Wp,q(X, Y ) and v ∈Wq ,p (Y ∗,X∗) (where p′ denotes the conjugate exponent of p), the map

t 7→ hu(t), v(t)iX,X∗ is absolutely continuous and

d ′ ′ hu(t), v(t)i ∗ = hu (t), v(t)i ∗ + hu(t), v (t)i ∗ for almost all t ∈ [0,T ]. (2.5) dt X,X Y,Y X,X

5 Proof. Observe that Y ∗ ֒→ X∗. By the density claim in (2.4), take a sequence u ∈ C1([0,T ]; X) such that ′ n′ p,q 1 ∗ q ,p ∗ ∗ un → u in W (X, Y ) and a sequence vn ∈ C ([0,T ]; Y ) with vn → v in W (Y ,X ) satisfying (since 1 ∗ vn ∈ C ([0,T ]; X ) too), by the previous lemma,

T T ′ ∗ ′ ∗ ′ ∗ ϕ (t)hun(t), vn(t)iX,X = − (hun(t), vn(t)iX,X + hun(t), vn(t)iX,X ) ϕ(t) Z0 Z0 T ′ ∗ ′ ∗ = − (hun(t), vn(t)iY,Y + hun(t), vn(t)iX,X ) ϕ(t) Z0 for all ϕ ∈ D(0,T ). It is trivial to pass to the limit (for the left-hand side, we use the fact that vn → v in C0([0,T ]; X∗) by the continuous embedding (2.3)) and we obtain the desired statement in (2.5) the sense of weak derivatives. The absolute continuity follows since the derivative is integrable. The lemma below provides a characterisation of the weak time derivative in terms of vector-valued test functions, which is the type of result that we wish to generate for the evolving case. Lemma 2.6 (Equivalent deﬁnition of weak time derivative). Let X ֒→ Y . The function u ∈ L1(0,T ; X) has a weak time derivative u′ ∈ L1(0,T ; Y ) if and only if

T T ′ ′ ∗ hu(t), η (t)iX,X∗ = − hu (t), η(t)iY,Y ∗ ∀η ∈ D((0,T ); Y ). (2.6) Z0 Z0 Proof. If u ∈ W1,1(X, Y ), then picking v = η ∈ D((0,T ); Y ∗) in Proposition 2.5 and integrating over time the formula for the derivative of the duality product gives (2.6). For the reverse direction, take η(t)= ϕ(t)f in (2.6) where ϕ ∈ D(0,T ) and f ∈ Y ∗ and we get

T T ′ ′ hϕ (t)u(t),fiX,X∗ = − hϕ(t)u (t),fiY,Y ∗ , Z0 Z0 and bringing the integral inside the duality product, we ﬁnd

T ϕ′(t)u(t)+ ϕ(t)u′(t),f =0 0 *Z +Y,Y ∗ because X ⊂ Y and f ∈ Y ∗. This implies (due to a corollary of the Hahn–Banach theorem) the ﬁrst deﬁnition.

:Remark 2.7. In the context of Remark 2.1, with the embedding ι: X ֒→ Y , the above lemma can be stated as the function u ∈ L1(0,T ; X) has a weak time derivative u′ ∈ L1(0,T ; Y ) if and only if

T T T ′ ∗ ′ ′ ∗ hι(t)u(t), η (t)iY,Y ∗ ≡ hu(t), ι(t) η (t)iX, X∗ = − hu (t), η(t)iY,Y ∗ ∀η ∈ D((0,T ); Y ). Z0 Z0 Z0 2.3 Gelfand triples The notion of Gelfand triples [48, §7.2], [12, §5.2] [58, Chapter III, §17], being omnipresent in applications in PDEs and in later sections of this paper, is worth discussing in detail. Let X be a reﬂexive Banach space, H d d ,∗be a Hilbert space and suppose that X ֒−→ H is a continuous and dense embedding. It follows that H∗ ֒−→ X and identifying H with its dual space through the Riesz map, we write the Gelfand triple

d .∗X ֒−→ H ֒→ X To be precise, let i: X → H be the canonical inclusion map, R: H → H∗ be the Riesz map and i∗ : H∗ → X∗ be the adjoint of i. Take f ∈ H. For any v ∈ X, by simply using the deﬁnition of the above maps, it follows that

∗ hi Rf, viX∗,X = hRf,iviH∗,H = (f,iv)H = (f, v)H .

The Gelfand triple convention is that we simply write the left-hand side above as hf, viX∗,X , i.e., we do not write i∗Rf since we have identiﬁed H and H∗. Furthermore, let us point out the following. • If X is a Hilbert space itself, the above Gelfand triple formula may appear to imply that X is a proper subset of its dual space X∗, seemingly contradicting the fact that X and X∗ are isomorphic. But this is not so: the map given by the composition of the canonical maps in X ֒→ H ֒→ H∗ ֒→ X∗ as described above (which necessarily involves the Riesz map from H to its dual) is not the same as the Riesz map RX from X to its dual X∗; the latter is deﬁned with the inner product of X.

6 • We represent functionals in X∗ with the inner product of H (and not the inner product of X in case X is a Hilbert space). More precisely, the duality pairing on X∗ ×X can be viewed as the continuous extension ∗ of (·, ·)H : H × X → R. Indeed, it can be shown [37, Theorem 2.9.2] that for every f ∈ X , there exists a sequence {fn}⊂ H such that (fn, x)H → hf, xiX∗,X for all x ∈ X. .If X is not reﬂexive, the embedding H∗ ֒→ X∗ may not be dense • Now, in the case of a Gelfand triple as described, we can write Proposition 2.5 as follows, which is a well known result (see [10, Theorem II.5.12]).

′ ′ Proposition 2.8. Let p ∈ [1, ∞]. For u ∈Wp,q(X,X∗) and v ∈Wq ,p (X,X∗), the map

t 7→ (u(t), v(t))H is absolutely continuous and

d ′ ′ (u(t), v(t)) = hu (t), v(t)i ∗ + hu(t), v (t)i ∗ for almost all t ∈ [0,T ]. dt H X ,X X,X

′ .(In particular, we have the continuous embedding Wp,p (X,X∗) ֒→ C0 ([0,T ]; H

p 3 Time-evolving Bochner spaces LX The aim in this section is to define a generalisation of the Bochner spaces Lp(0,T ; X) to describe integrable (in time) functions with values in a Banach space that itself depends on time. In [1, 3], two of the present authors p defined and studied properties of spaces LX given a sufficiently smooth parametrised family of Banach spaces {X(t)}t∈[0,T ]. These spaces were generalisations to the abstract Banach space setting of spaces introduced by Vierling in [56] in the context of Sobolev spaces on evolving surfaces. We now recall (and in some places, refine) the theory in [1] so that the presentation is essentially self-contained. For each t ∈ [0,T ], let X(t) be a real Banach space with X0 := X(0) and let

φt : X0 → X(t) be a bounded, linear, invertible map with inverse

φ−t : X(t) → X0.

It follows that the inverse is also bounded. These maps ‘link’ the time-dependent spaces and we call φt the pushforward map and φ−t the pullback map. We assume these satisfy the following properties. Assumption 3.1 (Compatibility). Suppose that

(1) φ0 is the identity,

(2) there exists a constant CX independent of t ∈ [0,T ] such that

kφ uk ≤ C kuk ∀u ∈ X , t X(t) X X0 0 kφ uk ≤ C kuk ∀u ∈ X(t), −t X0 X X(t)

(3) for all u ∈ X0, the map t 7→ kφtukX(t) is measurable.

We say that the pair (X(t), φt)t is compatible.

In what follows, we always assume that (X(t), φt)t satisﬁes Assumption 3.1 and we (formally) identify the family {X(t)}t∈[0,T ] with the symbol X.

Remark 3.2. Under this compatibility assumption, note that for s,t ∈ [0,T ], the map U(t,s) := φsφ−t : X(t) → X(s) defines a 2-parameter semigroup in the sense of [47, Definition 1.1.1]. Let us define the disjoint union

XT := X(t) ×{t}. t∈[[0,T ]

7 p Deﬁnition 3.3 (The space LX ). For p ∈ [1, ∞], deﬁne the space

p p LX := u : [0,T ] → XT , t 7→ (ˆu(t),t) | φ−(·)uˆ(·) ∈ L (0,T ; X0) . Identifying u(t)=(ˆu(t),t) with uˆ(t), endow the space with the norm

1 T p p 0 ku(t)kX(t) for p ∈ [1, ∞), kukLp := X  essR supt∈[0,T ] ku(t)kX(t) for p = ∞.

p 2 Theorem 3.4. Under Assumption 3.1, LX is a Banach space. If X is a family of Hilbert spaces, then LX is a Hilbert space with the canonical inner product

T (u, v) 2 := (u(t), v(t)) . LX X(t) Z0 p p Furthermore, L (0,T ; X0) and LX are isomorphic via φ(·) with an equivalence of norms:

1 p p p kukL ≤ φ−(·)u(·) Lp(0,T ;X ) ≤ CX kukL for all u ∈ LX . (3.1) CX X 0 X

Proof. For the ﬁrst two claims, see [3, Theorem 2.8] for the Hilbertian case and the paragraph after Deﬁnition 2.1 in [1] for the general Banach setting. The equivalence of norms is proved in [1, Lemma 2.3].

Spaces of smooth functions. The following Ck-type spaces will also be of use later. We start by deﬁning N k for k ∈ ∪{0}, the spaces CX of k-times continuously diﬀerentiable functions (on the closed interval [0,T ])

k k CX = η : [0,T ] → XT , t 7→ (η(t),t) | φ−(·)η(·) ∈ C ([0,T ]; X0) .

We will also need the space D X of smooth, compactly supported functions (but now on the open interval (0,T ))

DX = η : [0,T ] → XT , t 7→ (η(t),t) | φ−(·)η(·) ∈ D((0,T ); X0) . 3.1 Dual spaces

p In this section, we study the dual space of LX for appropriate p. First, we shall see that given a compatible p ∗ pair (X(t), φt)t∈[0,T ], we can also deﬁne the space LX∗ associated to the family {X (t)} by using dual maps. Indeed, denote by ∗ ∗ ∗ φ−t : X0 → X (t) the dual operator of φ−t : X(t) → X0. Under the condition

∗ ∗ t 7→ φ−tf X∗(t) is measurable for all f ∈ X0 , (3.2)

∗ ∗ it is not difficult to verify that the pair (X (t), φ−t)t∈[0,T ] is also compatible in the sense of the definition above (see [1, Remark 2.4]). This justifies the next definition.

p Deﬁnition 3.5 (The space LX∗ ). Given a compatible pair (X(t), φt)t∈[0,T ], under (3.2), we deﬁne the space p ∗ ∗ ∗ ∗ LX∗ using the dual spaces {X (t)}t∈[0,T ] and the dual maps {φ−(·) : X0 → X (t)}.

Remark 3.6. Note that if X0 is separable, then so is X(t) for every t ∈ [0,T ] and the condition (3.2) is satisfied. Regarding the relationship between the dual of Bochner spaces and Bochner spaces of the dual, recall that if Z is a reflexive Banach space, then Z∗ is also reflexive and hence it possesses the Radon–Nikodym property, ′ which is key to identifying the dual of Lp(0,T ; Z) as Lp (0,T ; Z∗) whenever p 6= ∞.

′ p p Theorem 3.7 (Identification of the dual of LX with LX∗ ). Suppose that the family of reflexive Banach spaces p ∗ {X(t)}t∈[0,T ] satisfies Assumption 3.1, (3.2) holds and let p ∈ [1, ∞). The dual space (LX ) is isometrically ′ p isomorphic to LX∗ (taken as in Definition 3.5) with duality pairing

T hf,ui p′ p = hf(t),u(t)iX∗(t),X(t). LX∗ ,LX Z0 p Furthermore, if p ∈ (1, ∞), then LX is reﬂexive.

8 Proof. The proof follows the classical proof for the corresponding result for Bochner spaces [20, §IV] with modiﬁcations, see Theorem 2.5 in [1]. We now establish a version of the fundamental theorem of calculus of variations for the evolving space setting. The proof is simple but we provide it to illustrate the kind of argument required when working with these kinds of spaces.

1 Lemma 3.8. If u ∈ LX is such that

T hu(t), η(t)iX(t),X∗(t) =0 ∀η ∈ DX∗ , Z0 then u ≡ 0.

∗ ∗ ∗ Proof. By writing u(t), η(t) ∗ = φ−tu(t), φt η(t) ∗ and setting ϕ := φ(·)η(·) ∈ D((0,T ); X0 ) it X(t),X (t) X0,X0 ∗ follows by the arbitrariness of η ∈ DX that T ∗ ∗ φ−tu(t), ϕ(t)iX0 ,X0 =0 ∀ϕ ∈ D((0,T ); X0 ), Z0 and we conclude with the help of Lemma 2.2 that φ−(·)u(·) ≡ 0, hence u ≡ 0. Remark 3.9 (Relation between the Riesz maps in the Hilbert space case). Suppose that {H(t)} is a family of Hilbert spaces compatible with a family of maps {φt} as above. Let us discuss the relationship between the various Riesz isomorphisms that are present in this situation. 2 2 ∗ 2 Let R: LH → LH∗ and St : H(t) → H (t) be the associated Riesz maps. If u, v ∈ LH , by deﬁnition t 7→ hStu(t), v(t)iH∗ (t),H(t) = (u(t), v(t))H(t) is measurable and we have

T T (u, v) 2 = (u(t), v(t)) = hS u(t), v(t)i ∗ LH H(t) t H (t),H(t) Z0 Z0 but on the other hand, by Theorem 3.7,

T (u, v) 2 = hRu, vi 2 2 = h(Ru)(t), v(t)i ∗ . LH LH∗ ,LH H (t),H(t) Z0 This implies that 2 Ru = S(·)u(·) in LH∗ ∗ ∗ 2 and thus (Ru)(t) = Stu(t) ∈ H (t) for almost all t. This suggests that identifying H(t) with H(t) forces LH 2 to be identiﬁed with LH∗ and vice versa.

3.2 Relation to measurable fields of Hilbert space As a digression, it is interesting to relate the notions introduced at the start of this section to the well established study of measurable fields of Hilbert spaces; see [21, Part II, Chapter 1], [52, Definition 8.9], [26, Appendix D] and references therein for more details. Definition 3.10. Let (Ω, F,µ) be a measurable space. A measurable field of Hilbert spaces is a family of Hilbert 1 spaces {H(ω)}ω∈Ω together with a subspace

M ⊆ H(ω) ωY∈Ω satisfying the properties

(i) the map ω 7→ kϕ(ω)kH(ω) is measurable for every ϕ ∈ M

(ii) if f ∈ ω H(ω) is such that ω 7→ (f(ω), ϕ(ω))H(ω) is measurable for every ϕ ∈ M, then f ∈ M

(iii) there existsQ a sequence {ϕn}⊂ M such that {ϕn(ω)}⊂ H(ω) is dense for all ω ∈ Ω.

1Recall that

H(ω) = f : Ω → H(ω) | f(ω) ∈ H(ω) Y  [  ω∈Ω ω∈Ω  

9 If only the first two conditions are satisfied, this defines an almost measurable field of Hilbert spaces. A measurable section ω 7→ f(ω) ∈ H(ω) is a called an Lp section if

p kf(ω)kH(ω) dµ(ω) < ∞. ZΩ The direct integral of H, denoted by L H(ω) dµ(ω), ZΩ is the Hilbert space containing all L2 sections of H under the inner product

(f,g)= (f(ω),g(ω))H(ω) dµ(ω). ZΩ We see then that in our setting of a compatible pair (H(t), φt)t∈[0,T ], Ω := [0,T ], F is a Borel σ-algebra on [0,T ] and µ is the Lebesgue measure, and our spaces {X(t)} in the case where they are Hilbert spaces form an almost measurable ﬁeld of Hilbert spaces. Furthermore, if X0 is separable, the word ‘almost’ in the previous 2 sentence can be omitted. The space LX corresponds to the direct integral of X.

4 Time derivatives in evolving spaces

Having defined Bochner-type spaces to deal with evolving families of Banach spaces, we focus in this section on defining a notion of a weak time derivative for functions in such spaces. Firstly, since the pullbacks of functions 1 in CX (recall the definition in §3) are differentiable, we are able to define a time derivative for such functions with a simple and natural formula. 1 0 Definition 4.1. A function u ∈ CX has a strong time derivativeu ˙ ∈ CX defined by X X ′ u˙(t)= φt (φ−tu) . (4.1) X • Evidently, this time derivative depends on the maps{φt }. We will sometimes also use the notation ∂ u in place ofu ˙. A similar definition could be stated for higher order derivatives but we will not need it in this text. 1 X Remark 4.2. This definition implies the following simple transport property: if u ∈ CX is of the form u = φt η for some η ∈ X0, then u˙ =0. The aim now is to look for a weaker notion of time derivative than the strong time derivative. Taking a cue from the integration by parts formula, we expect the definition of the weak time derivative to be similar to the non-moving setting but in view of the fact that the spaces here are evolving, we expect an additional term in its definition. Such a weak time derivative was defined in the setting of Hilbert triples X(t) ⊂ H(t) ⊂ X∗(t) (with each space a Hilbert space) in [3]. Here, we aim to drop the assumption of an existing pivot Hilbert space and define the weak time derivative in the full generality of the classical Banach space setting (see (2.6)). For the rest of this section, we work under the following assumptions: Assumption 4.3. We fix families X Y X(t), φt : X0 → X(t) t∈[0,T ] and Y (t), φt : Y0 → Y (t) t∈[0,T ], where X0 := X(0) and Y0 := Y (0), satisfying Assumption 3.1 and such that X(t)֒→ Y (t) continuously for all t ∈ [0,T ]. X Y Remark 4.4. We do not assume that φt = φt |X0 ! Doing so would lead to a simplified setting in what follows, see Remark 4.7 (ii) for more details.

4.1 Deﬁnition and properties of the weak time derivative

p q For a function u ∈ LX , we wish to deﬁne an appropriate concept of a weak time derivativeu ˙ ∈ LY motivated by the formula (2.6) in the non-moving setting, we take a test function η ∈ DY ∗ , and similar to the transport formula (2.5), we expect d hu(t), η(t)i ∗ = hu˙(t), η(t)i ∗ + hu(t), η˙(t)i ∗ + extra term, (4.2) dt X(t), X (t) Y (t),Y (t) X(t), X (t) where the extra term accounts for the time-dependence of the duality pairing. Integrating over [0,T ], and using the fact that η is compactly supported, this would lead to a weak derivative formula of the type (2.6). It remains for us to identify the extra term resulting from the derivative of the duality pairing and to justify this rigorously. To isolate the eﬀect of time-dependency that the evolution of the spaces induces in the associated duality product, we make the following assumption.

10 Assumption 4.5. We assume that (i) the map X Y ∗ t 7→ φt u0, (φ−t) v0 X(t),X∗(t) ∗ is continuously diﬀerentiable for ﬁxed u0 ∈ X0, v0 ∈ Y0 ; (ii) for all t ∈ [0,T ], the map

∗ ∂ X Y ∗ X × Y ∋ (u , v ) 7→ φ u , (φ ) v ∗ 0 0 0 0 ∂t t 0 −t 0 X(t),X (t) is continuous;

∗ (iii) there exists C > 0 such that, for almost all t ∈ [0,T ] and u0 ∈ X0, v0 ∈ Y0 , ∂ X Y ∗ ∗ φ u0, (φ ) v0 ∗ ≤ C ku0kX kv0kY . ∂t t −t X(t),X (t) 0 0

∗ ∗ × (Here, we have used the fact that Y (t) ֒→ X (t) continuously. It is convenient to deﬁne λ(t; ·, ·): X(t Y ∗(t) → R by ∂ X Y ∗ λ(t; u, v) := φt u0, (φ−t) v0 X(t),X∗(t) . (4.3) ∂t X Y ∗ (u0,v0)=(φ−tu,(φt ) v)

This leads us to the following generalization of the weak time derivative for functions that take values in evolving

Banach spaces. 1 Definition 4.6 (Weak time derivative). We say u ∈ LX is weakly differentiable with weak time derivative 1 v ∈ LY if T T T hu(t), η˙(t)iX(t),X∗(t) dt = − hv(t), η(t)iY (t),Y ∗(t) dt − λ(t; u(t), η(t)) dt ∀η ∈ DY ∗ . (4.4) Z0 Z0 Z0 In §7.1, we will see that this definition recovers the well-established definition of the weak material derivative in the Gelfand triple setting where the pivot space is an L2-type space on an evolving domain or surface. We note that this generalises to the fully Banach space case the definition in the work [3] co-authored by the first and final authors where all spaces were assumed to be Hilbert spaces in a Gelfand triple setting. Remark 4.7. (i) The first two parts of Assumption 4.5 imply that λ is a Carathéodory function, thus for 1 1 u ∈ LX and v ∈ LY ∗ , the superposition map t 7→ λ(t; u(t), v(t)) is measurable. (ii) The expression for λ suggests that our definition could lead to problems in the case Y (t) := X(t) with the Y X same maps φt ≡ φt , in which case λ ≡ 0 and the extra term in the definition of a weak time derivative 1 would vanish. But this is indeed the case for smooth functions u ∈ CX . To wit, omitting the exponent in X ∗ φt = φt , we have, for any η ∈ DX (0,T ), T T ′ hu˙(t), η(t)iX(t),X∗(t) = hφt(φ−tu(t)) , η(t)iX(t),X∗(t) Z0 Z0 T ∗ ′ ∗ = − hφ−tu(t), (φt η(t)) iX0,X0 (by (2.6)) Z0 T ∗ ∗ ′ = − hu(t), φ−t(φt η(t)) iX(t),X∗(t) Z0 T = − hu(t), η˙(t)iX(t),X∗(t). Z0 p Hence, our setting includes the case Y (t) ≡ X(t) and the calculation above shows that u ∈ LX is weakly X differentiable (in the sense of (4.4)) if and only if φ−tu is weakly differentiable (in the sense of (2.6)), X X ′ and it holds that u˙ = φt (φ−tu) . (iii) Note that the above is different to the case where there is a Hilbert triple framework in place and the q derivative has sufficient smoothness to lie in LX : in such a case, we would still get a non-zero λ term! That is, p q {u ∈ LX :u ˙ ∈ LX } and ∗ p q q (X(t),H(t),X (t)) is a Gelfand triple; {u ∈ LX :u ˙ ∈ LX∗ ∩ LX } are fundamentally different since the derivative (·˙) in each set is a different operator; in particular, the second is defined through the pivot space. One should take care to not confuse the two.

11 By a simple application of Lemma 3.8, we can prove the next result.

1 1 Proposition 4.8 (Uniqueness of weak derivatives). Suppose u ∈ LX has weak time derivatives v1, v2 ∈ LY . Then v1 = v2.

1 0 Proposition 4.9 (Strong derivatives are also weak derivatives). Let u ∈ CX and u˙ ∈ CX be its strong time derivative. Then u is also weakly differentiable with weak time derivative u˙. We provide the proof later on page 14 since we will need an additional result to prove it. Remark 4.10. Proposition 4.9 shows that our notion of a weak derivative is indeed a generalisation of the strong derivative (4.1). It would perhaps seem more natural to define the weak derivative by pulling back to the X Y reference domain with the maps φ−t, differentiating in the usual (weak) sense, and pushing forward with φt . Even though this is the case when Y (t) ≡ X(t) (as per Remark 4.7), this approach does not lead to the same definition as above when the spaces do not coincide. On this topic, note further that:

p X (i) If u ∈ LX is weakly diﬀerentiable in the sense we deﬁned above, it is not necessarily the case that φ−tu has a weak time derivative (in the usual sense). Conditions under which this is true will be explored in §4.6.

p X (ii) Even if u ∈ LX is such that φ−tu is weakly diﬀerentiable, then a simple calculation shows that the function Y X ′ φt (φ−tu) does not satisfy an expression of the form (4.4) unless Y (t) ≡ X(t). Indeed, it is easy to check that

T T Y X ′ Y X ∗ φt (φ−tu) , η Y (t),Y ∗(t) = − φt φ−tu, η˙ Y (t),Y ∗(t) ∀η ∈ DY . Z0 Z0

4.2 Transport formula for smooth functions and further remarks Having deﬁned a notion of weak time derivative, we now demonstrate that a transport formula of the form (4.2) holds for suﬃciently smooth functions.

1 1 ∗ Lemma 4.11. Let Assumption 4.5 hold. Given σ1 ∈ CX , σ2 ∈ CY ∗ , the map t 7→ hσ1(t), σ2(t)iX(t),X (t) is absolutely continuous and for almost all t ∈ [0,T ], d hσ (t), σ (t)i ∗ = hσ˙ (t), σ (t)i ∗ + hσ (t), σ˙ (t)i ∗ + λ(t; σ (t), σ (t)). dt 1 2 X(t),X (t) 1 2 X(t),X (t) 1 2 X(t),X (t) 1 2 For the proof, it becomes convenient to introduce the following notation and definitions. Definition 4.12. For t ∈ [0,T ], we define the following objects: (i) the evolution of the duality pairing, ∗ R X Y ∗ X Y ∗ π(t; ·, ·): X0 × Y0 → , π(t; u, v) := φt u, (φ−t) v Y (t),Y ∗(t) = φt u, (φ−t) v X(t),X∗(t); ∂ λˆ(t; ·, ·): X × Y ∗ → R, λˆ(t; u, v) := π(t; u, v); 0 0 ∂t

(ii) the map Πt : X0 → Y0 deﬁned by Y X Πtu := φ−t φt u, which satisﬁes

Πtu, v ∗ = π(t; u, v). Y0,Y0

ˆ ∗∗ (iii) the map Λ(t): X0 → Y0 deﬁned by

ˆ ∗∗ ∗ ˆ hΛ(t)u0, v0iY0 ,Y0 := λ(t; u0, v0).

∗ The fact that π(t; ·, ·) is defined over X0 × Y0 is motivated by the discussion preceding the definition of the weak time derivative above, allowing for the formulation in (4.4) with test functions in DY ∗ . We see that λ, defined in (4.3) is the pushforward of the bilinear form λˆ:

ˆ X Y ∗ λ(t; u, v)= λ(t; φ−tu, (φt ) v).

For convenience, let us write Assumption 4.5 in terms of the notation of π and λˆ.

12 Remark 4.13. Assumption 4.5 is equivalent to the following:

∗ (i) the map t 7→ π(t; u, v) is continuously diﬀerentiable for ﬁxed u ∈ X0, v ∈ Y0 with derivative ∂ λˆ(t; ·, ·): X × Y ∗ → R, λˆ(t; u, v) := π(t; u, v); 0 0 ∂t

(ii) for all t ∈ [0,T ], the map (u, v) 7→ λˆ(t; u, v) is continuous;

∗ (iii) there exists C > 0 such that, for almost all t ∈ [0,T ] and u ∈ X0, v ∈ Y0 ,

ˆ ∗ |λ(t; u, v)|≤ C kukX0 kvkY0 .

With this, we obtain that Λˆ has a dual operator

ˆ ∗ ∗∗∗ ∗ Λ(t) : Y0 → X0 .

p It is worthwhile noting that for u ∈ L (0,T ; X0), the map t 7→ Πtu(t) is measurable from (0,T ) → Y0 since X p p Y p p φ(·)u(·) ∈ LX ⊂ LY and by deﬁnition of compatibility and (3.1), φ−(·) : LY → L (0,T ; Y0), ensuring measurability of the composition map. Remark 4.14 (The Gelfand triple case). Some observations regarding the deﬁnition above are timely.

∗ Y X ∗ (i) Consider Y (t) := X (t) with maps φt = (φ−t) , and that there exists a family of Hilbert spaces H(t) d H H X such that X(t) ֒−→ H(t). We suppose H(t) evolves with maps φt satisfying φt |X0 = φt and that we have a Gelfand triple structure X(t) ֒→ H(t) ֒→ X∗(t). In this case, the deﬁnition of the operator π above becomes, for u ∈ X0, v ∈ X0,

X X H H H H π(t; u, v)= hφt u, φt viX(t),X∗(t) = hφt u, φt viX(t),X∗(t) = (φt u, φt v)H(t),

and this deﬁnition can be uniquely extended to H0 × H0 by density of X0 in H0. This also shows that the map Πt satisﬁes

∗ H A X Πt : X0 → H0 ⊂ X0 , Πtu = (φt ) φt u,

A where (·) stands for the Hilbert adjoint. We can extend the previous deﬁnition to H0 as the operator (still labelled as Πt)

∗ H A H Πt : H0 → H0 ⊂ X0 , Πtu = (φt ) φt u.

In particular, when X(t) is also a Hilbert space, we recover2 the deﬁnitions in [3].

(ii) In the setting above, observe that the deﬁnition of the operator π, and consequently of Πt and λˆ, can be expressed involving the ﬂows and inner product solely of the intermediate Hilbert space H(t), and as such all of these are independent of the base space X(t) that is chosen.

1 1 ∗ Proof of Lemma 4.11. Let us ﬁrst show that givenσ ˆ1 ∈ C ([0,T ]; X0) andσ ˆ2 ∈ C ([0,T ]; Y0 ), the map t 7→ 1 π(t;ˆσ1(t), σˆ2(t)) is C ([0,T ]) and that for all t ∈ [0,T ], d π(t;ˆσ (t), σˆ (t)) = π(t;ˆσ′ (t), σˆ (t)) + π(t;ˆσ (t), σˆ′ (t)) + λˆ(t;ˆσ (t), σˆ (t)). (4.5) dt 1 2 1 2 1 2 1 2 To see this, start by considering for any h> 0 the diﬀerence quotient

π(t + h;ˆσ (t + h), σˆ (t + h)) − π(t;ˆσ (t + h), σˆ (t + h)) δ π(t;ˆσ (t), σˆ (t)) = 1 2 1 2 h 1 2 h + π (t; δhσˆ1(t), σˆ2(t + h)) + π (t; σ1(t),δhσˆ2(t)) .

The continuity of π with respect to the second and third variables and the regularity of σˆ1,σ ˆ2 implies that, for all t ∈ [0,T ] and as h → 0, the sum of the last two terms on the right-hand side above converges to

′ ′ π(t;ˆσ1(t), σˆ2(t)) + π(t;ˆσ1(t), σˆ2(t)).

2 In [3], the notations Tt and ˆb were used in place of Πt and π respectively.

13 We now use Assumption 4.5 (or equivalently, the conditions in Remark 4.13) to establish that for almost all t ∈ [0,T ], π(t + h;ˆσ (t + h), σˆ (t + h)) − π(t;ˆσ (t + h), σˆ (t + h)) ∂π 1 2 1 2 → (t;ˆσ (t), σˆ (t)) = λˆ(t; σ (t), σ (t)). h ∂t 1 2 1 2 Indeed, let us fix t ∈ [0,T ) and h > 0 be sufficiently small so that t + h ≤ T . We have, using the absolute continuity of s 7→ π(s; u, v) for fixed u and v, π(t + h;ˆσ (t + h), σˆ (t + h)) − π(t;ˆσ (t + h), σˆ (t + h)) 1 2 1 2 − λˆ(t; σ (t), σ (t)) = h 1 2 1 t+h = λˆ(s;ˆσ (t + h), σˆ (t + h)) ds − λˆ(t;ˆσ (t), σˆ (t)) h 1 2 1 2 Zt =I+II+III, where we have set 1 t+h T I := λˆ(s;ˆσ (t + h), σˆ (t + h)) − λˆ(s;ˆσ (t), σˆ (t + h)) ds = λˆ (s; δ σˆ (t), σˆ (t + h)) χ (s) ds, h 1 2 1 2 h 1 2 [t,t+h] Zt Z0

1 t+h T II := λˆ(s;ˆσ (t), σˆ (t + h)) − λˆ(s;ˆσ (t), σˆ (t)) ds = λˆ (s;ˆσ (t),δ σˆ (t)) χ (s) ds, h 1 2 1 2 1 h 2 [t,t+h] Zt Z0 and 1 t+h III := λˆ(s;ˆσ (t), σˆ (t)) ds − λˆ(t;ˆσ (t), σˆ (t)). h 1 2 1 2 Zt Now observe that, for sufficiently small h, the integrands in I and II are uniformly bounded and converge pointwise to 0, and so the Dominated Convergence Theorem implies that I, II → 0 as h → 0. Since for fixed u, v the map s 7→ λˆ(s; u, v) is integrable, it follows from Lebesgue’s Differentiation Theorem that also III → 0 1 as h → 0 for all t ∈ [0,T ] proving that t 7→ π(t; σ1(t), σ2(t)) has a continuous derivative and is thus C ([0,T ]). We can reason similarly for t ∈ (0,T ] and h> 0 with t − h ≥ 0, and this will show (4.5). From here, the claimed X Y ∗ statement can be obtained directly by takingσ ˆ1(t) := φ−tσ1(t) andσ ˆ2(t) := φt σ2(t). With this transport formula at hand, we can prove our earlier claim that strong derivatives are also weak derivatives.

0 0 1 ∗ Proof of Proposition 4.9. We start by observing thatu ˙ ∈ CX ⊂ CY ⊂ LY . Given η ∈ DY , we have T T X X ′ hu˙(t), η(t)iY (t),Y ∗(t) = hφt (φ−tu(t)) , η(t)iY (t),Y ∗(t) Z0 Z0 T X ′ Y ∗ = π(t; (φ−tu(t)) , (φt ) η(t)) Z0 T T X Y ∗ ′ ˆ X Y ∗ = − π(t; φ−tu(t), (φt ) η(t) ) − λ(t; φ−tu(t), (φt ) η(t)) (by Lemma 4.11) Z0 Z0 T T = − hu(t), η˙(t)iX(t),X∗(t) − λ(t; u(t), η(t)), Z0 Z0 which proves the claim.

4.3 A characterisation of the weak time derivative We come now to an alternative characterisation of the weak time derivative related to the derivative of a duality product, which turns out to be useful in various situations (eg. in the mechanics of applying the Galerkin method for existence of solutions to nonlinear PDEs, see §8), cf. [53, Lemma 1.1, §III] for the non-moving ∗∗ case. First, let us introduce some notation. For a Banach space Z, we denote by JZ : Z → Z the (linear and bounded) canonical injection into the double dual:

∗ hJZ u,fiZ∗∗, Z∗ := hf,uiZ∗, Z , ∀f ∈ Z , u ∈ Z. In part to avoid working with double and triple duals, it sometimes becomes useful to assume that ˆ Λ(t)u ∈ Range(JY0 ), ∀t ∈ [0,T ], u ∈ X0. (4.6)

14 Remark 4.15. Regarding the assumption (4.6), note that

• it is automatically satisﬁed if Y0 is reﬂexive; • the meaning of the assumption is that

∗∗ ∗ ∗ ˆ ∗∗ ∗ ∗ ∀u ∈ X0, ∃y ∈ Y0 : hJY0 y,fiY0 ,Y0 = hf,yiY0 ,Y0 = hΛ(t)u,fiY0 ,Y0 ∀f ∈ Y0 ; (4.7)

• denoting the map u 7→ y in (4.7) by y = Lu, we can write

∗ ˆ ∗∗ ∗ hf,LuiY0 ,Y0 = hΛ(t)u,fiY0 ,Y0 ,

which suggests that Λ(ˆ t) can be identiﬁed as a map Λ(ˆ t): X0 → Y0 and this is indeed what we shall do below whenever the assumption is in force.

p q Proposition 4.16 (Characterisation of the weak time derivative). Assume (4.6). Let u ∈ LX and g ∈ LY . Then u˙ = g if and only if

d Y ∗ Y ∗ Y ∗ ∗ hu(t), (φ ) vi ∗ = hg(t), (φ ) vi ∗ + hΛ(t)u(t), (φ ) vi ∗ ∀v ∈ Y . (4.8) dt −t X(t),X (t) −t Y (t),Y (t) −t Y (t),Y (t) 0

Y ∗ ∗ Proof. Making the substitution η = (φ−(·)) v for arbitrary v ∈ Y0 in (4.8), we ﬁnd

T T ′ ψ (t)hu(t), η(t)iX(t),X∗(t) = − ψ(t) hg(t), η(t)iY (t),Y ∗(t) + hΛ(t)u(t), η(t)iY (t),Y ∗(t) ∀ψ ∈ D(0,T ). Z0 Z0 Collecting terms, we may write this as

T ′ 0= hψ (t)u(t)+ ψ(t)(g(t)+Λ(t)u(t)), η(t)iY (t),Y ∗(t) Z0 T ′ Y Y ∗ = hψ (t)φ−tu(t)+ ψ(t)φ−t(g(t)+Λ(t)u(t)), viY0,Y0 . Z0 Bringing the integral inside the ﬁrst part of the duality pairing above, we get d φY u(t)= φY (g(t)+Λ(t)u(t)). dt −t −t

Y 1 Now, as φ−(·)(g +Λu) ∈ L (0,T ; Y0), by the characterisation in Lemma 2.6 of the weak derivative through vector-valued test functions, this is equivalent to

T T Y ′ ∗ Y ∗ ∗ hφ−tu(t), ξ (t)iY0,Y0 = − hφ−t(g(t)+Λ(t)u(t)), ξ(t)iY0 ,Y0 ∀ξ ∈ D((0,T ); Y0 ). Z0 Z0

Y ∗ ∗ Y ∗ ′ Setting ϕ := (φ−(·)) ξ ∈ DY so thatϕ ˙ = (φ−(·)) ξ , we can pushforward the duality products above to obtain

T T T hu(t), ϕ˙(t)iY (t),Y ∗(t) = − hg(t), ϕ(t)iY (t),Y ∗(t) − λ(t; u(t), ϕ(t)). Z0 Z0 Z0

This being valid for every ϕ ∈ DY ∗ shows thatu ˙ = g by deﬁnition. The reverse implication follows since every step in the above proof is an equivalence.

4.4 Evolving Sobolev–Bochner spaces Having deﬁned an appropriate notion of weak time derivative, we consider in this section the deﬁnition and properties of evolving Sobolev–Bochner spaces, which are the spaces in which solutions to parabolic PDEs (on evolving spaces) typically lie in. These can be considered to be the time-evolving versions of Wp,q(X, Y ) (introduced in (2.2)). To reiterate, we again are enforcing Assumption 4.3.

Deﬁnition 4.17 (The space Wp,q(X, Y )). For p, q ∈ [1, ∞], deﬁne the space

p,q p q W (X, Y ) := {u ∈ L | u˙ ∈ L } with norm kukWp,q := kuk p + ku˙k q . X Y (X,Y ) LX LY Proposition 4.18. The space Wp,q(X, Y ) is a Banach space.

15 Wp,q p Proof. Let {un} be a Cauchy sequence in (X, Y ). It follows that un → u in LX to some u andu ˙ n → w in q ∗ LY to some w. We have for all η ∈ DY ,

T T T hu˙ n(t), η(t)iY (t),Y ∗(t) = − hun(t), η˙(t)iX(t),X∗(t) − λ(t; un(t), η(t)). Z0 Z0 Z0 It is immediate to pass to the limit in the ﬁrst two terms, and for the last one we observe that, since λ is bilinear,

T T ′ ∞ λ(t; u (t), η(t)) − λ(t; u(t), η(t)) ≤ C kηk ∗ ku − uk 1 → 0. n LY n LX 0 0 Z Z

We then have T T T hw(t), η(t)iY (t),Y ∗(t) = − hu(t), η˙(t)iX(t),X∗(t) − λ(t; u(t), η(t)), Z0 Z0 Z0 which shows, by uniqueness of weak derivatives (Proposition 4.8), that w =u ˙.

p p In Theorem 3.4, we saw that φ(·) acts as an isomorphism between the spaces L (0,T ; X0) and LX with an equivalence of norms. A natural question to ask is: under which conditions does φ(·) act as an isomorphism between W(X0, Y0) and W(X, Y ) with an equivalence of norms? This question will be addressed in a later section. First, let us formalise this idea and give a simple density result under such an equivalence.

p,q p,q Deﬁnition 4.19. We say there is an evolving space equivalence between W (X, Y ) and W (X0, Y0) if Wp,q X p,q v ∈ (X, Y ) if and only if φ−(·)v(·) ∈W (X0, Y0), and the following equivalence of norms holds:

X X C1 φ v(·) ≤kvkWp,q ≤ C2 φ v(·) . −(·) p,q (X,Y ) −(·) p,q W (X0,Y0) W (X0,Y0)

We may also say that Wp,q(X, Y ) has the evolving space equivalence property or that Wp,q(X, Y ) and p,q W (X0, Y0) are equivalent instead of ‘evolving space equivalence’. This notion of an evolving space equivalence is important as it ensures that properties of the classical p,q p,q spaces W (X0, Y0) carry over to the time-dependent W (X, Y ). As mentioned, we investigate when such an equivalence exists in §4.6. For now, we prove the following useful results, which are direct generalisations of (2.3) and (2.4).

p,q p,q Lemma 4.20. Suppose that there exists an evolving space equivalence between W (X0, Y0) and W (X, Y ). Wp,q 0 .i) The embedding (X, Y ) ֒→ CY is continuous) 1 Wp,q (ii) The space CX is dense in (X, Y ). Proof. The statement in (i) is a consequence of the following series of implications:

Wp,q X p,q X 0 0 u ∈ (X, Y ) ⇐⇒ φ−(·)u(·) ∈W (X0, Y0)=⇒ φ−(·)u(·) ∈ C ([0,T ]; Y0) ⇐⇒ u ∈ CY , where we used (2.3) for the middle implication. Wp,q X p,q To prove (ii), let u ∈ (X, Y ), so that v(·) := φ−(·)u(·) ∈W (X0, Y0). According to (2.4), we can take a 1 p,q X 1 sequence (vn)n ⊂ C ([0,T ]; X0) such that vn → v in W (X0, Y0) as n → ∞. Deﬁning un(·) := φ(·)vn(·) ∈ CX , we have, due to the evolving space equivalence,

X X kun − ukWp,q(X,Y ) ≤ C kφ−(·)un − φ−(·)ukWp,q(X,Y ) = C kvn − vkWp,q (X,Y ) → 0 as n → ∞.

4.5 Diﬀerentiating the duality product: transport theorem In this section we state and prove a transport theorem for general functions in the abstract spaces deﬁned above in the vein of Proposition 2.5. Theorem 4.21 (Transport theorem). Let either

′ ′ (i) p ∈ [2, ∞], u ∈ Wp,p (X, Y ) and v ∈ Wp,p (Y ∗,X∗) or

16 ′ ′ (i’) p ∈ [1, ∞], u ∈ Wp,p(X, Y ) and v ∈ Wp ,p (Y ∗,X∗), and suppose that in either case the spaces involved have the evolving space equivalence property. Then the map

t 7→ u(t), v(t) X(t), X∗(t) (4.9) is absolutely continuous and we have, for almost all t ∈ [0 ,T ], d hu(t), v(t)i ∗ = u˙(t), v(t) ∗ + u(t), v˙(t)i ∗ + λ(t; u(t), v(t)). (4.10) dt X(t),X (t) Y (t),Y (t) X(t), X (t)

Proof. Under either of the assumptions it follows that both (4.9) and the right-hand side of (4.10) deﬁne functions in L1(0,T ). This is clear for case (i’), and in case (i) simply observe that p′ ≤ 2 ≤ p, and thus ′ p p u ∈ LX ⊂ LX , so (4.9) and the last term in (4.10) are also integrable. It therefore remains to prove that the right hand side of (4.10) is the weak derivative of (4.9). But this 1 1 follows by density. Indeed, take sequences {un}n ⊂ CX , {vn}n ⊂ CY ∗ such that

′ ′ p,p p,p ∗ ∗ un → u in W (X, Y ) and vn → v in W (Y ,X ).

We then have, using Lemma 4.11, d hu (t), v (t)i ∗ = hu˙ (t), v (t)i ∗ + hu (t), v˙ (t)i ∗ + λ(t; u (t), v (t)). dt n n X(t),X (t) n n Y (t),Y (t) n n X(t),X (t) n n Writing this in terms of the deﬁnition of the weak derivative and then passing to the limit as in the proof of Proposition 2.5, we ﬁnd that (4.10) holds in the weak sense, giving the conclusion. Remark 4.22. Let us motivate the conditions on the exponents in the statement above. Assume that u ∈ Wp1,q1 (X, Y ) and v ∈ Wp2,q2 (Y ∗,X∗) and suppose that these spaces have the evolving space equivalence property. The displayed equations in Theorem 4.21 above reveal that conditions on the exponents are necessary:

• (4.9) must deﬁne an integrable function, but this is the case for any exponents p1, q1,p2, q2, due to the 0 extra regularity v ∈ CX∗ ; • the right-hand side of (4.10) must also be integrable:

′ ′ – the ﬁrst and second terms show that we must have Lp2 ⊂ Lq1 and Lp1 ⊂ Lq2 ; ′ p p – the last term requires L 1 ⊂ L 2 . This shows that the extra term λ(t; u(t), v(t)) — which is not present in the classical setting — holds us back ′ from stating a general result for u, v ∈ Wp,p , though in some applications we can ﬁnd a way around this obstacle (as we will see in Sections 5 and 7).

4.6 Criteria for evolving space equivalence Here, we focus on obtaining conditions that can be checked ensuring an evolving space equivalence (see Def- p,q p,q inition 4.19) between W (X0, Y0) and W (X, Y ). The main result is the next theorem which states the precise conditions required; the reader is also referred to Theorem 5.6 for the statement (and proof) of this theorem applied to the particular case of a Gelfand triple (the setting of which results in some simpliﬁcations in the conditions that are needed). We recall the operators and bilinear forms in Deﬁnition 4.12 and we write M(0,T ; Z) to stand for the set of Bochner measurable maps f : (0,T ) → Z into a Banach space Z. Theorem 4.23 (Criteria for evolving space equivalence). Let Assumption 4.5 hold, let p, q ∈ [1, ∞] and suppose that for all t ∈ [0,T ], the range condition (4.6) and the following hold:

Πt : X0 → Y0 has a linear extension Πt : Y0 → Y0 which is bounded uniformly in t, (4.11)

Π(·)u ∈M(0,T ; Y0) for all u ∈ X0, (4.12) −1 Πt : Y0 → Y0 exists and is uniformly bounded in t, (4.13) −1 Π(·) v ∈M(0,T ; Y0) for all v ∈ Y0, (4.14) † −1 p,q ∗ ∗ p, p∧q ∗ ∗ † −1 ∗ −1 (Π ) : W (Y0 , Y0 ) →W (Y0 , Y0 ) where ((Π ) f)(t) := (Πt ) f(t). (4.15)

p,q p,q Then there is an evolving space equivalence between W (X0, Y0) and W (X, Y ). More precisely,

17 p,q X Wp,q (i) under (4.11) and (4.12), if u ∈W (X0, Y0), then φ(·)u ∈ (X, Y ) and

• X Y ′ ∂ φt u(t)= φt Πtu (t).

Wp,q X p,q (ii) under (4.6) and (4.11)–(4.15), if u ∈ (X, Y ), then φ−(·)u ∈W (X0, Y0) and

X ′ −1 Y φ−tu(t) = Πt φ−tu˙(t). Remark 4.24. Regarding these assumptions, let us make the following observations.

d i) If X0 ֒−→ Y0, then (4.12) follows immediately from (4.11). Indeed, for any y ∈ Y0, take a sequence xn ∈ X0) with xn → y in Y0. As Πtxn = Πtxn → Πty in Y0 and the pointwise limit of measurable functions is measurable, the claim holds.

† −1 p ∗ p ∗ (ii) Assumptions (4.13) and (4.14) imply that (Π ) : L (0,T ; Y0 ) → L (0,T ; Y0 ) is a bounded linear operator.

(iii) Assumption (4.15) is analogous to the assumption on the diﬀerentiability of Πt (or πt).

(iv) One should bear in mind that Πt has an inverse only on the set Πt(X0) (i.e. its range):

X φY | φt −t X(t) Y X0 −→ X(t) −→ φ−t X(t) ⊂ Y0.

It is not clear that Πt(X0) is closed (and hence not necessarily a Hilbert space in its own right) so the −1 dual of Πt is not well deﬁned in general. This is why we only talk about the inverses of Π and its dual operator. The rest of this section is dedicated to proving this result, which will be done in a number of steps. We begin with some preliminaries: we have the pointwise dual maps

∗ ∗ ∗ ∗ ∗ ∗ Πt : Y0 → X0 and Πt : Y0 → Y0

∗ and it is not diﬃcult to see that for every f ∈ Y0 , ∗ ∗ ∗ ∗ hΠt f, xiY0 ,Y0 = hΠt f, xiX0 ,X0 whenever x ∈ X0. (4.16) Let us construct the Nemytskii operators

† ∗ † ∗ (Πu)(t) := Πtu(t), (Πu)(t)= Πtu(t), (Π f)(t) = Πt f(t), (Π f)(t)= Πt f(t). We have that p p p p Π: L (0,T ; X0) → L (0,T ; Y0), Π: L (0,T ; Y0) → L (0,T ; Y0), † p ∗ p ∗ † p ∗ p ∗ Π : L (0,T ; Y0 ) → L (0,T ; X0 ), Π : L (0,T ; Y0 ) → L (0,T ; Y0 ), are all bounded and linear for any p: for the first two maps this follows respectively by definition (see §4) and by (4.11) and (4.12)3, and the latter two because the dual of a bounded linear operator is also bounded and linear with the same operator norm. Note carefully that Π† is in general not the same as Π∗ (which is defined † as the dual of Π) since we are not in the reflexive setting (and likewise for Π )! See the next remark for more on this. Remark 4.25. (i) Regarding Π and Π, we know that their dual operators satisfy by definition, for any p ∈ [1, ∞],

∗ p ∗ p ∗ ∗ p ∗ p ∗ Π : L (0,T ; Y0) → L (0,T ; X0) and Π : L (0,T ; Y0) → L (0,T ; Y0) .

If it were the case that we could identify the duals of the above Bochner spaces with the expected spaces (which we can do for example in the reﬂexive setting for appropriate exponents p), then the above can be written as

∗ p ∗ p ∗ ∗ p ∗ p ∗ Π : L (0,T ; Y0 ) → L (0,T ; X0 ) and Π : L (0,T ; Y0 ) → L (0,T ; Y0 ), and it is easy to see in this case that

† ∗ Π† ≡ Π∗ and Π ≡ Π . 3 The measurability of the image of the latter operator follows because Π(·)(·) is by assumption a Carath´eodory function.

18 † (ii) Assumptions (4.13) and (4.14) imply that Π has an inverse given by

† −1 −1 † −1 † −1 ∗ ∗ −1 (Π ) ≡ (Π ) where ((Π ) f)(t) := (Πt ) f(t) = (Πt ) f(t), and furthermore, both maps

−1 p p † −1 −1 † p ∗ p ∗ Π : L (0,T ; Y0) → L (0,T ; Y0) and (Π ) ≡ (Π ) : L (0,T ; Y0 ) → L (0,T ; Y0 ) are bounded linear operators. † The next proposition shows that Π and Π take differentiable functions into differentiable functions thanks to the assumptions on the differentiability of π that were made earlier. Even though one does not usually distinguish between an element of a Banach space and its action as an element of the corresponding double dual space, in the proofs below, to emphasise that we do not assume reflexivity of neither X0 nor Y0, we will always ∗ write explicitly the canonical injections JX0 , JY0 , JY0 . Proposition 4.26 (Differentiability of Πu). Let p, q ∈ [1, ∞] and suppose that (4.11) and (4.12) hold. If p,q u ∈W (X0, Y0), then Πu satisfies

T T ′ ′ ∗∗ ∗ ∗ ∗ ∗ ˆ hΠtu(t), ϕ (t)iY ,Y = − Πtu (t), ϕ(t) + hΛ(t)u(t), ϕ(t)iY0 ,Y0 ∀ϕ ∈ D((0,T ); Y0 ). 0 0 Y0,Y0 Z0 Z0

ˆ p, p∧q In particular, if Λ(t)u(t) ∈ Range(JY0 ), then Πu ∈W (Y0, Y0) with (Πu)′(t)= Π u′(t)+ J −1Λ(ˆ t)u(t). (4.17) t Y0 Here, p ∧ q := min(p, q).

∗ 1 Proof. Let us take ϕ ∈ D((0,T ); Y0 ) and u ∈ C ([0,T ]; X0) to obtain, by (4.5) in the proof of Lemma 4.11,

T T ′ ′ hϕ (t), Πtu(t)i ∗ = π(t; u(t), ϕ (t)) Y0 ,Y0 Z0 Z0 T d = π(t; u(t), ϕ(t)) − π(t; u′(t), ϕ(t)) − λˆ(t; u(t), ϕ(t)) dt Z0 T ′ ∗ ˆ = − hΠtu (t), ϕ(t)iY ,Y + Λ(t)u(t), ϕ(t) ∗∗ ∗ 0 0 Y ,Y Z0 0 0 T D E ′ ∗ ˆ = − Πtu (t), ϕ(t) + Λ(t)u(t), ϕ(t) ∗∗ ∗ . Y0,Y0 Y ,Y Z0 0 0 D E ˆ If Λu is in the range of JY0 , then we can write the last term above as

T J −1Λ(ˆ t)u(t), ϕ(t) , Y0 ∗ 0 Y0,Y0 Z D E p∧q 1 proving (4.17) (with the right-hand side belonging to L (0,T ; Y0)) for u ∈ C ([0,T ]; X0) due to the character- 1 isation of the weak time derivative of Lemma 2.6. The conclusion now follows from the density of C ([0,T ]; X0) 1,1 in W (X0, Y0) and the continuity of Π, Π and Λ.ˆ Note that the assumption that was needed for (4.17) above is exactly (4.6). Now we look for a converse of Proposition 4.26. In order to do so, we need a preparatory result in the form of the next lemma. † Lemma 4.27 (Diﬀerentiability of Π v). Let p, q ∈ [1, ∞] and suppose that (4.11) and (4.12) hold. If v ∈ p,q ∗ ∗ † p, p∧q ∗ ∗ W (Y0 , Y0 ), then Π v ∈W (Y0 ,X0 ) with

† ∗ ′ ′ ˆ ∗ ∗ (Π v) (t)= Πt v (t)+ Λ(t) JY0 v(t).

† p, p∧q ∗ ∗ Proof. We will ﬁrst prove the intermediary result that Π v ∈W (X0 ,X0 ) with ∗ † ′ ∗ ′ ′ ˆ ∗ ∗ ∗ (Π v) (t) = (Πt v(t)) = Πt v (t)+ Λ(t) JY0 v(t) in X0 (4.18) 1 ∗ for v taken as stated in the lemma. Indeed, approximating with v ∈ C ([0,T ]; Y0 ) and denoting a test function by ϕ ∈ D((0,T ); X0), we have

T T ∗ ′ ′ hΠ v(t), ϕ (t)i ∗ = π(t; ϕ (t), v(t)) t X0 , X0 Z0 Z0

19 T = − π(t; ϕ(t), v′(t)) + λˆ(t; ϕ(t), v(t)) Z0 T ′ ∗ ˆ = − hΠtϕ(t), v (t)iY ,Y + Λ(t)ϕ(t), v(t) ∗∗ ∗ 0 0 Y ,Y Z0 0 0 T D E ′ ∗ ˆ ∗ = − hΠtϕ(t), v (t)iY ,Y + Λ(t)ϕ(t), JY v(t) ∗∗ ∗∗∗ 0 0 0 Y ,Y Z0 0 0 T D E ∗ ′ ˆ ∗ ∗ = − ϕ(t), Πt v (t)+ Λ(t) JY0 v(t) ∗ . (4.19) 0 X0,X0 Z D E Take ϕ(t)= ψ(t)x where ψ ∈ D(0,T ) and x ∈ X0; this becomes

T T ′ ∗ ∗ ′ ∗ ψ (t) hΠ v(t), xi ∗ = − ψ(t) x, Π v (t)+ Λ(ˆ t) JY ∗ v(t) . (4.20) t X0 , X0 t 0 ∗ 0 0 X0,X0 Z Z D E Manipulating and pulling the integrals inside the duality pairing, we get

T ′ ∗ ∗ ′ ˆ ∗ ∗ ψ (t)Πt v(t)+ ψ(t)(Πt v (t)+ Λ(t) JY0 v(t)), x =0. * 0 + ∗ Z X0 , X0

Since this is true for every x ∈ X0, this gives, by deﬁnition of the weak time derivative,

∗ ′ ∗ ′ ˆ ∗ ∗ (Πt v(t)) = Πt v (t)+ Λ(t) JY0 v(t)

∗ ∗ and here, using the identity (4.16) relating Πt and Πt as well as a density argument for v, we deduce that (4.18) p,q ∗ ∗ is satisﬁed for each v ∈W (Y0 , Y0 ). 1 ∗ Now, let us conclude. Again with v ∈ C ([0,T ]; Y0 ) and a test function ϕ ∈ D((0,T ); X0), we calculate

T T ∗ ′ ∗ ′ ∗ Πt v(t), ϕ (t) ∗ = hΠt v(t), ϕ (t)iX ,X (using (4.16)) X ,X 0 0 Z0 0 0 Z0 D E T ∗ ′ ˆ ∗ ∗ = − ϕ(t), Πt v (t)+ Λ(t) JY v(t) , 0 X , X∗ Z0 0 0 D (from (4.19E ) and using (4.16) again) and from here we follow the same argument elucidated above (beginning with the derivation of (4.20)) and this † p,q ∗ ∗ † p ∗ will show that Π v ∈W (X0 ,X0 ). Since we already know that Π v ∈ L (0,T ; Y0 ), the claim follows. We are now ready to provide a converse to Proposition 4.26. Proposition 4.28 (“Diﬀerentiability of Π−1v”). Let p, q ∈ [1, ∞]. Suppose (4.6) and (4.11)–(4.15) hold. If p p, p∧q p,q u ∈ L (0,T ; X0) is such that Πu ∈W (Y0, Y0), then u ∈W (X0, Y0) with

−1 −1 u′ = Π (Πu)′ − Π J −1Λˆu. Y0

∗ † −1 p,p∧q ∗ ∗ Proof. Let ϕ ∈ D((0,T ); Y0 ) and deﬁne v := (Π ) ϕ. By(4.15), v ∈W (Y0 , Y0 ) and we can apply Lemma 4.27 to get (noting that p ∧ (p ∧ q)= p ∧ q)

† ′ ′ ˆ ∗ ∗ p∧q ∗ ϕ = Π v + Λ JY0 v in L (X0 ). Taking u as stated, noting that

ˆ ∗ ∗ ∗ ˆ ∗ ∗∗ ∗∗∗ hu(t), Λ (t)JY0 v(t)iX0 ,X0 = hΛ(t)u(t), JY0 v(t)iY0 ,Y0

ˆ ∗∗ ∗ = hΛ(t)u(t), v(t)iY0 ,Y0

ˆ ∗ = hΛ(t)u(t), v(t)iY0 ,Y0 (with the ﬁnal equality as explained in Remark 4.15), we ﬁnd

T T ∗ ′ ∗ ′ ˆ ∗ ∗ ∗ hu(t), ϕ (t)iX0 ,X0 = hu(t), Πt v (t)+ Λ JY0 v(t)iX0 ,X0 Z0 Z0 T ′ ∗ ˆ ∗ = hΠtu(t), v (t)iY0,Y0 + hΛ(t)u(t), v(t)iY0 ,Y0 (because u(t) ∈ X0) Z0

20 T ′ ∗ ˆ ∗ = − h(Πtu(t)) , v(t)iY0,Y0 − hΛ(t)u(t), v(t)iY0 ,Y0 0 Z (by integration by parts, see below) T −1 ∗ −1 ∗ ′ ∗ ˆ ∗ = − hΠt (Πtu(t)) , Πt v(t)iY0 ,Y0 − hΠt Λ(t)u(t), Πt v(t)iY0,Y0 Z0 T −1 −1 ′ ∗ ˆ ∗ = − hΠt (Πtu(t)) , ϕ(t)iY0,Y0 − hΠt Λ(t)u(t), ϕ(t)iY0 ,Y0 . Z0 Assumptions (4.13), (4.14) and the assumptions on Λˆ imply that

−1 ′ q −1 p Π ((Πu) ) ∈ L (0,T ; Y0) and Π Λˆu ∈ L (0,T ; Y0)

p, p∧q and thus u ∈W (X0, Y0) as desired. In the above manipulation, we used Lemma 2.5 (with X and Y in the notation of that lemma chosen as X := Y := Y0 ) to ﬁrst obtain

d ′ ′ hΠtu(t), v(t)iY ,Y ∗ = h(Πtu(t)) , v(t)iY ,Y ∗ + hΠtu(t), v (t)iY ,Y ∗ , dt 0 0 0 0 0 0 which upon integrating and using the compact support of ϕ leads to

T T ′ ∗ ′ ∗ h(Πtu(t)) , v(t)iY0 ,Y0 = − hΠtu(t), v (t)iY0,Y0 . Z0 Z0 Finally, we are able to prove the main result.

p,q X p Proof of Theorem 4.23. Suppose u ∈W (X0, Y0), then immediately φ(·)u(·) ∈ LX , so it remains to prove that q ∗ this function has a weak time derivative in LY . Let η ∈ DY , then

T T X X Y ∗ Y ∗ ′ φt u(t), η˙(t) X(t), X∗(t) = φt u(t), (φ−t) (φt ) η(t) Y (t),Y ∗(t) Z0 Z0 T Y ∗ ′ = Πtu(t), (φt ) η(t) ∗ Y0,Y0 Z0 T ∗ ∗ ′ Y ˆ Y ∗∗ ∗ = − Πtu (t), φt η(t) + hΛ(t)u(t), φt η(t)iY ,Y Y ,Y ∗ 0 0 Z0 0 0 D E (by Proposition 4.26) T T Y ′ X = − φt Πtu (t) , η(t) Y (t),Y ∗(t) − λ(t; φt u(t), η(t)), Z0 Z0 X from where we conclude that t 7→ φt u(t) has a weak time derivative as desired. p,q For the converse direction, we begin by ﬁxing u ∈ W (X, Y ). By deﬁnition, for any η ∈ DY ∗ ,

T T T

u˙(t), η(t) Y (t),Y ∗(t) = − u(t), η˙(t) X(t),X∗(t) − λ(t; u(t), η(t)), Z0 Z0 Z0 which we can pullback, arguing as in the previous paragraph and rearrange to obtain

T T Y Y ∗ ˆ X Y ∗ X Y ∗ ′ φ−tu˙(t), φt η(t) ∗ + Λ(t)φ−tu(t), φt η(t) ∗∗ ∗ = − Πt φ−tu(t) , φt η(t) ∗ . 0 Y0,Y0 Y0 ,Y0 0 Y0,Y0 Z D E D E Z D E ∗ Y ∗ Letting ϕ := φ(·) η ∈ D((0,T ); Y0 ) and using assumption (4.6), this is equivalent to T T Y −1 X X ′ ˆ ∗ φ−tu˙(t)+ JY Λ(t)φ−tu(t), ϕ(t) ∗ = − Πt φ−tu(t) , ϕ (t) , 0 Y ,Y Y0,Y0 Z0 0 0 Z0 D E from where we conclude that

′ Π φX u(t) = φY u˙(t)+ J −1Λ(ˆ t)φX u(t) t −t −t Y0 −t

X p,p∧q X p,q with Πφ−(·)u ∈W (X0, Y0). By Proposition 4.28 it now follows that φ−(·)u ∈W (X0, Y0). The equivalence of norms is a result of the uniform boundedness of the ﬂow maps and their inverses, and from the assumptions on Λˆ and Π.

21 5 The Gelfand triple X(t) ⊂ H(t) ⊂ X∗(t) setting

We now specialise the theory and results of §4 to the important case of a Gelfand triple (recall §2.3)

d (X(t) ֒−→ H(t) ֒→ X∗(t for all t ∈ [0,T ], that is, X(t) is a reﬂexive Banach space continuously and densely embedded into a Hilbert space H(t) which has been identiﬁed with its dual via the Riesz map. This setup arises frequently in the study of evolutionary variational problems and several concrete examples will be given in §7 and §8. In the context of §4, we are taking Y (t) := X∗(t) with the inclusion of X(t) into Y (t) given through compositions of the maps involved in the Gelfand triple. Naturally, we wish to make use of the theory developed in the previous sections and the basic assumptions that one needs (namely, Assumption 4.5) translated into this Gelfand triple framework are as follows. Assumption 5.1. For all t ∈ [0,T ], assume the existence of maps H X H φt : H0 → H(t), φt := φt |X0 : X0 → X(t) such that H X (H(t), φt )t∈[0,T ] and (X(t), φt )t∈[0,T ] are compatible pairs. We assume the measurability condition (3.2), i.e., ∗ ∗ t 7→ φ−tf X∗(t) is measurable for all f ∈ X0 .

Furthermore, suppose that (i) for fixed u ∈ H0, H 2 t 7→ kφt ukH(t) is continuously differentiable; (ii) for fixed t ∈ [0,T ], ∂ (u, v) 7→ (φH u, φH v) is continuous, ∂t t t H(t) and there exists C > 0 such that, for almost all t ∈ [0,T ] and for any u, v ∈ H0, ∂ (φH u, φH v) ≤ Ckuk kvk . (5.1) ∂t t t H(t) H0 H0

It follows that ∗ X ∗ (X (t), (φ−t) )t∈[0,T ] is a compatible pair. ˆ ∗ Under the final assumption above, the map Λ(t): X0 → X0 , defined in Definition 4.12, is in fact such that ˆ ∗ Λ(t): H0 → H0 is bounded and linear with

ˆ ∗ ˆ hΛ(t)u, viH0 ,H0 = λ(t; u, v) ∀u, v ∈ H0.

Remark 5.2. Parts (i) and (ii) of Assumption 5.1 say that, with4 H A H H H Πt = (φt ) φt , π(t; u, v) = (φt u, φt v)H(t), (5.2) the map (u, v) 7→ λˆ(t; u, v) = ∂π(t; u, v)/∂t is continuous and there exists C > 0 such that, for almost all t ∈ [0,T ] and for any u, v ∈ H0, ˆ |λ(t; u, v)|≤ CkukH0 kvkH0 . Taking into view the Hilbert structure, the deﬁnition of the weak time derivative in (4.4) becomes the following. 1 1 Deﬁnition 5.3 (Weak time derivative). We say u ∈ LX has a weak time derivative v ∈ LX∗ if T T T (u(t), η˙(t))H(t) dt = − hv(t), η(t)iX∗ (t),X(t) dt − λ(t; u(t), η(t)) dt ∀η ∈ DX . Z0 Z0 Z0 It is convenient to state Proposition 4.16 applied to this setting. p q Proposition 5.4 (Characterisation of the weak time derivative). Let u ∈ LX and g ∈ LX∗ . Then u˙ = g if and only if d H X H (u(t), φ v) = hg(t), φ vi ∗ + λ(t; u(t), φ v) ∀v ∈ X . dt t H(t) t X (t),X(t) t 0 4See Remark 4.14 (i).

22 5.1 Diﬀerentiating the inner product: transport theorem W ∗ 0 We now specialise Theorem 4.21 to this setting. We ﬁrst obtain the extra regularity (X,X ) ֒→ CH as a consequence of the evolving space equivalence property, and then use it to obtain a general statement. Theorem 5.5 (Transport theorem in the Gelfand triple setting). Let p ∈ [1, ∞] and suppose that there exists ′ ′ p,p ∗ Wp,p ∗ an evolving space equivalence between W (X0,X0 ) and (X,X ). Then ′ Wp,p ∗ 0 ;i) the embedding (X,X ) ֒→ CH is continuous) ′ (ii) given u, v ∈ Wp,p (X,X∗), the map

t 7→ (u(t), v(t))H(t) (5.3)

is absolutely continuous and we have, for almost all t ∈ [0,T ], d (u(t), v(t)) = u˙(t), v(t) ∗ + u(t), v˙(t)i ∗ + λ(t; u(t), v(t)). (5.4) dt H(t) X (t),X(t) X(t),X (t)

Proof. The proof of (i) follows from

′ ′ Wp,p ∗ X p,p ∗ X 0 u ∈ (X,X ) ⇐⇒ φ−(·)u ∈W (X0,X0 )=⇒ φ−(·)u ∈ C ([0,T ]; H0) ⇐⇒ u ∈ CH ,

H X where we have used the assumption, Theorem 2.8 and the fact that φt |X0 = φt . We now turn to the proof of (ii). The fact that (5.3) is an element of L1(0,T ) and that (5.4) is the weak time derivative of (5.3) follows as in the proof of Theorem 4.21, so it suﬃces now to check that the right-hand side of (5.4) is also in L1(0,T ). Due to (i) and the stronger assumption (5.1) we may conclude with

T T |λ(t; u(t), v(t))| dt ≤ C ku(t)k kv(t)k dt ≤ CT kuk 0 kvk 0 . H(t) H(t) CH CH Z0 Z0 ∗ Let us now study criteria for the spaces W(X,X ) and W(X0, Y0) to be equivalent like in §4.6.

5.2 Criteria for evolving space equivalence The evolving space equivalence criteria of Theorem 4.23 tailored to the situation under consideration is as follows. It is worth pointing out that these conditions are considerably easier to check in practice than the ones given in [3, Theorem 2.33]. Theorem 5.6 (Criteria for evolving space equivalence in the Gelfand triple setting). Let Assumption 5.1 hold. If for all t ∈ [0,T ],

Πt : X0 → X0 is bounded uniformly in t, (5.5) −1 Πt : X0 → X0 exists and is bounded uniformly in t, (5.6) −1 Π(·) u ∈M(0,T ; X0) for all u ∈ X0, (5.7) −1 p,q p,p∧q Π : W (X0,X0) →W (X0,X0), (5.8)

Wp,q ∗ p,q ∗ then (X,X ) and W (X0,X0 ) are equivalent.

More precisely, with Πt as in (5.11), p,q ∗ X Wp,q ∗ (i) under (5.5), if u ∈W (X0,X0 ), then φ(·)u ∈ (X,X ) and

• X X ∗ ′ ∂ φt u(t) = (φ−t) Πtu (t). (5.9)

Wp,q ∗ X p,q ∗ (ii) under (5.5)–(5.8), if u ∈ (X,X ), then φ−(·)u ∈W (X0,X0 ) and

X ′ −1 X ∗ φ−tu(t) = Πt (φt ) u˙(t). (5.10) Proof. The idea is to verify the assumptions of Theorem 4.23. Since we are in the reﬂexive setting, assumption (4.6) is automatic and Remark 4.25 applies and we do not need to distinguish between Π† and Π∗. Assumption # ∗ ∗ (5.5) implies that the existence of the dual Πt : X0 → X0 to Πt considered as an operator Πt : X0 → X0 which is deﬁned (as usual) by # ∗ ∗ ∗ hΠt f, xiX0 ,X0 := hf, ΠtxiX0 ,X0 ∀f ∈ X0 , x ∈ X0.

23 Now, if f ∈ X0, the right-hand side equals (f, Πtx)H0 . On the other hand, because (5.5) is in force, by the ∗ self-adjoint property of Πt : X0 → X0 ,

∗ hΠtf, xiX0 ,X0 = (f, Πtx)H0 ∀f, x ∈ X0.

# This shows that Πt |X0 ≡ Πt and hence we may take as an extension (of Πt)

# Πt := Πt . (5.11)

∗ ∗ Observe that Πt : X0 → X0 is bounded uniformly in t because Πt : X0 → X0 is bounded uniformly by assumption and taking the dual preserves norms. This gives (4.11). The measurability assumption (4.12) follows by Remark 4.24. Let us now see that the inverse of Πt exists and that (4.13) is veriﬁed. Thanks to (5.6), we may deﬁne −1 # ∗ ∗ −1 (Πt ) : X0 → X0 as the dual of Πt : X0 → X0. We also see that, arguing as above,

−1 # −1 ∗ −1 (Πt ) |X0 = (Πt ) = Πt ,

−1 # −1 −1 # ∗ i.e., (Πt ) extends Πt . We claim that (Πt ) is indeed the inverse of Πt. To see this, take f ∈ X0 and a ∗ sequence xn ∈ X0 with xn → f in X0 . It follows that

−1 # −1 (Πt ) Πtxn = Πt Πtxn = xn → y

−1 # −1 # −1 but by continuity, the left-hand side converges to (Πt ) Πty, and hence we have shown that (Πt ) = (Πt) (in the sense of the left inverse, the right inverse follows by the same argument). The remaining claims in assumption (4.13) follow by the same reasoning as above. Assumption (4.14) on the measurability is implied by (5.7) and (again) a density argument just as in Remark 4.24. ∗ ∗ −1 −1 By reﬂexivity, it follows that Πt ≡ Πt and hence (Πt ) = Πt , so that (5.8) directly gives (4.15). The conclusion now follows from Theorem 4.23.

# Remark 5.7. It is important to emphasise that Πt := Πt deﬁned in the proof above is, in general, diﬀerent to ∗ ∗ Πt , the dual of Πt : X0 → X0 .

5.3 Alternative criteria for (5.8) For some applications, it may turn out that (5.8) (or (4.15)) is too cumbersome or inconvenient to verify in practice (as the case will be in one of the examples we consider below), so we would like to have alternative H A criteria to replace it. This is what we focus on now. Deﬁning ξt := (φ−t) , it follows that the pair (H(t), ξt)t∈[0,T ] is compatible if H A H0 is separable or t 7→ (φ−t) u H(t) is measurable for u ∈ H0. (5.12)

Lemma 5.8. Under (5.5), (5.6), (5.7), (5.12) and if for all t ∈ [0,T ],

Assumption 4.5 holds for the maps ξt; (5.13) ˆξ ∗ ˆ ξ the map Λ (t): X0 → X0 satisﬁes Λ (t)(X0) ⊂ X0; (5.14) −1 ∗ (Πt ) : X0 → X0 exists and is bounded uniformly in t, (5.15) then assumption (5.8) holds.

Proof. These assumptions allow us to apply the theory developed in this section now with the maps ξt. The # ∗ ∗ ∗ proof of Theorem 5.6 above shows that Πt := Πt : X0 → X0 is an extension of Πt to X0 . Likewise, the map ξ A −1 ξ −1 ∗ ∗ Π = ξt ξt = Πt : X0 → X0 has an extension Πt = (Πt ) to X0 , which by (5.15) satisﬁes (4.11), (4.12). We ξ −1 can, using (5.14), thus apply Proposition 4.26 to the map Π = Π (with the X0 and Y0 in the statement of the proposition chosen to be X0), which implies (5.8) with

−1 ′ −1 ∗ ′ ˆ ξ p,q (Πt u(t)) = (Πt ) u (t)+ Λ (t)u(t) ∀u ∈W (X0,X0).

−1 ∗ ∗ ∗ # −1 ∗ # −1 Remark 5.9. Note that the map (Πt ) : X0 → X0 relates to Πt via (Πt ) = (Πt ) .

24 5.4 Evolving space equivalence for the space W(X,H) q It can sometimes be the case that solutions to PDEs have the time derivative belonging not just to LX∗ but q W the more regular space LH . In this case, we say that solutions belong to (X,H) and it can be useful to know under which circumstances this space is equivalent to W(X0,H0). Theorem 5.10 (Criteria for regularity of evolving space equivalence in the Gelfand triple setting). Let the p,q p,q assumptions of Theorem 5.6 hold. Then W (X,H) and W (X0,H0) are equivalent. Proof. We ﬁrst need some basic properties of the various adjoint and dual maps. An easy calculation shows X ∗ H A that (φt ) |X(t) ≡ (φt ) : X(t) → H0 and hence, by density of X(t) ⊂ H(t),

X ∗ H A (φt ) |H(t) ≡ (φt ) . (5.16)

By the same reasoning,

X ∗ H A (φ−t) |H0 ≡ (φ−t) . (5.17)

p,q p,q ∗ • X X ∗ ′ Now, from the formula (5.9), for u ∈W (X0,H0) ⊂W (X0,X0 ), we have ∂ φt u(t) = (φ−t) Πtu (t). Since Πt = Πt on X0 and the right-hand side is well deﬁned and bounded from H0 into H0 (see (5.2)), we have ′ q Πt|H0 = Πt too. Thanks to this and utilising the additional regularity that u ∈ L (0,T ; H0), we get

• X X ∗ ′ X ∗ H A H ′ H ′ ∂ φt u(t) = (φ−t) Πtu (t) = (φ−t) (φt ) φt u (t)= φt u (t) where for the last equality we used (5.17). In the other direction, taking u ∈ Wp,q(X,H), now the formula (5.10) gives5

X ′ −1 X ∗ −1 H A −1 H A H A H −1 H A H (φ−tu(t)) = Πt (φt ) u˙(t)= Πt (φt ) u˙(t) = Πt (φt ) u˙(t) = ((φt ) φt ) (φt ) u˙(t)= φt u˙(t)

−1 where we made use of (5.16) and the fact that Πt can be deﬁned on H0 (just as we argued above). The proof reveals that a function u ∈ W(X,H) has a weak time derivative given by

H X ′ u˙(t)= φ−t(φ−tu(t)) which is a natural generalisation of the formula for the strong time derivative.

6 The Aubin–Lions lemma in evolving spaces

Our aim is to generalise the following result (see e.g. [48, Lemma 7.7]). Aubin–Lions lemma. Let X, Y and Z be Banach spaces such that X is separable and reﬂexive. Suppose c c ∞ >X ֒−→ Z is compact and Z ֒→ Y is injective. Then Wp,q(X, Y ) ֒−→ Lp(0,T ; Z) is also compact for any 1

5 H A p H A H Let us note that t 7→ (φt ) w(t) is measurable for every w ∈ LH from (0, T ) to H0 since (φt ) w(t)=Πtφ−tw(t) is measurable as remarked in §4.

25 Assumption 6.1. In addition to the compatible pairs

X Y X(t), φt : X0 → X(t) t∈[0,T ] and Y (t), φt : Y0 → Y (t) t∈[0,T ], with X(t) ⊂ Y (t) just as in §3, we assume the existence of an additional family of Banach spaces

Z {Z(t)}t∈[0,T ], φt : Z0 → Z(t) t∈[0,T ]

Z c Z X . such that (Z(t), φt )t∈[0,T ] is compatible and X0 ֒→ Z0 ֒→ Y0. We also assume φt |X0 = φt p,q p,q Theorem 6.2 (Aubin–Lions lemma). Under Assumption 6.1, suppose that W (X0, Y0) and W (X, Y ) are equivalent in the sense of Deﬁnition 4.19. For any p ∈ (1, ∞) and q ∈ [1, ∞], the embedding

Wp,q c p X, Y ) ֒−→ LZ) is compact. Wp,q X Proof. Suppose (un)n is a bounded sequence in (X, Y ), then by the equivalence of spaces (φ−(·)un(·))n is p,q p bounded in W (X0, Y0). By the classical Aubin–Lions lemma, it has a convergent subsequence in L (0,T ; Z0), Z Z X say (φ−(·)unk (·))k. But then, using the uniform boundedness of φ(·), (φ(·)φ−(·)unk )k = (unk )k also converges in p LZ , proving the result.

Part II: Applications

7 Examples of function spaces on evolving domains and surfaces

In the following examples we consider spaces of Lebesgue integrable or Sobolev functions over evolving domains and surfaces. We will prove that the theory of this paper can be applied to these cases, which should be useful when studying a wide variety of evolutionary problems on moving domains and surfaces. In particular, we will show that evolving space equivalences hold, which can be rather non-trivial.

Evolving domains and surfaces. Let us begin with the basic assumptions and notations that we need in order to describe evolving domains and surfaces. Assumption 7.1. We assume the following. (i) For each t ∈ [0,T ], let

M(t) be a bounded C2 domain Ω(t) ⊂ Rn or a C2 n-dimensional hypersurface Γ(t) ⊂ Rn+1,

with Ω(t) connected and Γ(t) compact (i.e., without boundary). (ii) Deﬁne

n if M(t)=Ω(t) d = (7.1) (n +1 if M(t)=Γ(t)

Let w : [0,T ] × Rd → Rd ∈ C0([0,T ], C2(Rd, Rd)) be a given vector field that we interpret to be a velocity 0 Rd Rd field. The evolution in time of {M(t)}t∈[0,T ] is related through a map Φ(·) : [0,T ] × → such that 0 2 Φt : M0 → M(t) is a C diffeomorphism satisfying the (Banach space) ODE d Φ0 = w(t, Φ0), dt t t 0 Φ0 = Id.

0 1 2 Rd Rd 0 It follows from the assumption above that Φ(·) ∈ C ([0,T ], C ( , )) and Φt (∂M0) = ∂M(t). Further- more we denote t 0 −1 Φ0 := (Φt ) . In the next sections, we study the following cases involving Gelfand triples: (i) H(t)= L2(M(t)) with X(t)= W 1,r(M(t))

26 (ii) H(t)= H1(M(t)) with X(t)= W 2,r(M(t)) (iii) H(t)= H−1(Ω(t)) with X(t)= Lp(Ω(t)) ∩ H−1(Ω(t)), and the non-Gelfand triple example

k,r 1 2,r (iv) X(t) = W (Γ(t)) with Y (t) = L (Γ(t)) (for k = 0 and k = 1) and X(t) = W0 (Ω(t)) with Y (t) = 1,1 W0 (Ω(t)). t We stress that these spaces are independent of the ﬂow map Φ0. Before we proceed, we need to introduce some more concepts and properties.

Pushforward and pullback maps. For functions u: M0 → R, we deﬁne the pushforward map φt by

t φtu := u ◦ Φ0. (7.2) 0 R Its inverse φ−tv = v ◦ Φt acting on functions v : M(t) → is called the pullback map.

Diﬀerential operators and integration by parts The notation g(t) will be used to refer to the Riemannian metric tensor associated to M(t) and ∇g(t) will stand for the usual gradient when M(t)=Ω(t) and the surface gradient (or tangential gradient) when M(t)=Γ(t); the latter can be seen as the projection of the gradient (of 0 a suitable extension) of the function onto the tangent space. We write DΦt for the Jacobian matrix of partial 0 derivatives of Φt (which, in case M(t) = Γ(t), refers to the tangential partial derivatives with respect to the ambient space). Note that, in either case, this denotes an (n + 1) × (n + 1) matrix. The integration by parts formula on surfaces [25, Theorem 2.10] for suﬃciently smooth functions is

u∂iv = − v∂iu + uvH(t)νi(t), ZΓ(t) ZΓ(t) ZΓ(t) where ∂i refers to the ith component of ∇g(t), ν(t) is the unit normal vector on Γ(t), and H(t) is the mean curvature of Γ(t) deﬁned as the sum of the principal curvatures. Deﬁning the determinant of the Jacobian matrix

0 0 Jt := detDΦt ,

0 from continuity and J0 = 1 we have its uniform boundedness: there exists a constant CJ > 0 such that

−1 0 0 < CJ ≤ Jt ≤ CJ ∀t ∈ [0,T ].

0 1 Moreover, from the regularity assumptions on the velocity ﬁeld, it follows J(·) ∈ C ([0,T ] ×M0) and

d J 0 = φ (∇ · w(t))J 0. (7.3) dt t −t g(t) t We also sometimes use the following transport formula (see [25, Equation (5.8)] in the case of an evolving surface):

d ∇ u · ∇ v = ∇ u˙ · ∇ v + ∇ u · ∇ v˙ + ∇ u⊺H(t)∇ v, (7.4) dt g(t) g(t) g(t) g(t) g(t) g(t) g(t) g(t) ZM(t) ZM(t) where the notation (·)⊺ means the transpose of the matrix and we deﬁned the deformation tensor

⊺ H := (∇g · w)Id − Dgw + (Dgw) .

We refer the reader to [4, 24, 25] and citations therein for full details on (evolving) hypersurfaces and their deﬁnitions in this context. For later use it is convenient to introduce the following positive-deﬁnite (with a constant that is uniform in time) matrix and its determinant

(DΦ0)⊺DΦ0 if M(t)=Ω(t), A0 t t 0 A0 t := 0 ⊺ 0 at := det t . ((Dg0 Φt ) Dg0 Φt + ν0 ⊗ ν0 if M(t)=Γ(t),

When M(t)=Γ(t), we have that (see Proposition 4.1 of [16])

0 −1 t t ⊺ (At ) = φ−t((Dg(t)Φ0)(Dg(t)Φ0) )+ ν0 ⊗ ν0.

27 Transformation of diﬀerential operators We record the following expressions (see [51, Proposition 2.29, Lemma 2.30, Lemma 2.62, Equation (2.91), p. 64] for the ﬂat case, and [16, Section 3] for surfaces):

t 0 −1 J0 = φt((Jt ) ), and, given suﬃciently smooth functions u: M(t) → R and v : M0 → R, we have: (i) for the gradient operator, via the chain rule for tangential gradients,

0 ⊺ ∇g0 (φ−tu) = (Dg0 Φt ) φ−t ∇g(t)u , and to invert the formula in the case of a surface we need again to add the term corresponding to the normal component, yielding

0 0 −1 φ−t ∇g(t)u = Dg0 Φt (At ) ∇g0 (φ−tu) , (7.5)

(ii) for the Laplace–Beltrami operator,

1 0 0 −1 φ−t(∆g(t)u)= ∇g0 · at (At ) ∇g0 φ−tu , (7.6) a0 t q p1 t t −1 φt(∆g0 v)= ∇g(t) · a (A0) ∇g(t)φtv . (7.7) at 0 0 q We now explore some particular examples. Wep refer to §7.5 for any proofs that are omitted (for reasons of clarity) in the next sections.

7.1 L2(M(t)) pivot space In this subsection we present the most commonly occurring case where the pivot space is an L2 space, namely H(t) := L2(M(t)). This example was already analysed (for M(t) = Γ(t) and various X(t)) in [4] but due to its importance and universal role in many applications, we will treat it afresh here for the convenience of the reader and for completeness. Let r ≥ 2 and deﬁne X(t) := W 1,r(M(t)) and Y (t) := X∗(t) = W 1,r(M(t))∗; for X(t), we take the usual norm 1/r r r kukW 1,r (M(t)) := |u| + |∇g(t)u| . ZM(t) ! Hence, we have the Gelfand triple structure W 1,r(M(t)) ⊂ L2(M(t)) ⊂ W 1,r(M(t))∗.

We denote by φt the pushforward map deﬁned above in (7.2). 2 1,r Lemma 7.2. Under Assumption 7.1, the pairs (L , φt)t∈[0,T ] and (W , φt)t∈[0,T ] are compatible. Since we have a Gelfand triple structure, by Remark 4.14, the evolution of the duality pairing has the form

0 π(t; u, v)= uvJt . ZM0 By simply diﬀerentiating and using the formula (7.3) for diﬀerentiating the determinant of the Jacobian and then pushing forward, we obtain

λ(t; u, v)= uv∇g(t) · w(t). ZM(t) 2 2 2 Deﬁnition 7.3 (L (M) weak time derivative). A function u ∈ LX has a weak time derivativeu ˙ ∈ LX∗ if and only if

T T T hu˙(t), η(t)iX∗t),X(t) = − u(t)η ˙(t) − u(t)η(t)∇g(t) · w(t) ∀η ∈ DX . Z0 Z0 ZM(t) Z0 ZM(t) We can then prove: Proposition 7.4. Under Assumption 7.1, given r ≥ 2 and for any p, q ∈ [1, ∞], there exists an evolving space p,q 1,r 1,r ∗ p,q 1,r 1,r ∗ equivalence between the spaces W (W (M0), W (M0) ) and W (W , (W ) ).

28 Applications. There are numerous examples of PDEs on evolving domains or surfaces with L2 as the pivot space. Some equations are analysed in [1, 4, 5], and here we mention a few of them. (1) The archetypal equation to consider (on a surface) is the surface advection-diﬀusion equation

u˙ − ∆gu + u∇g · w =0 onΓ(t),

u(0) = u0 on Γ0,

2 1 where u0 ∈ L (Γ0). In this case, X(t) = H (Γ(t)) and the evolving space equivalence and well posedness are proved in [4]. (2) Similar computations can be derived for systems of equations with bulk-surface interactions. Here we mention the coupled bulk-surface system that was studied in [4], in which case both Ω(t), Γ(t) ⊂ Rn+1 and we have Γ(t)= ∂Ω(t): u˙ − ∆Ωu + u∇Ω · w = f on Ω(t),

u˙ − ∆Γv + v∇Γ · w + ∇Ωu · ν = g on Γ(t),

∇Ωu · ν = βv − αu on Γ(t),

u(0) = u0 on Ω0,

v(0) = v0 on Γ0, 1 1 1 1 where u0 ∈ H (Ω0), v0 ∈ H (Γ0), α,β > 0 are given constants. Setting X(t) = H (Ω(t)) × H (Γ(t)) and H(t)= L2(Ω(t)) × L2(Γ(t)), one can show existence for the system (see [4, §5.3]). The analysis and properties of a more complicated and nonlinear coupled bulk-surface system can be found in [5]. (3) Moreover in [4, §5.4.1] the authors considered the fractional Sobolev space X(t) = H1/2(Γ(t)) and proved W ∗ ∗ that (X,X ) and W(X0,X0 ) are equivalent — a fact which was used to aid with the study of the fractional porous medium equation 1/2 m u˙ + (−∆g) (u )+ u∇g · w =0 onΓ(t),

u(0) = u0 on Γ(0),

∞ m m−1 1/2 in [2]. Here, m ≥ 1, u0 ∈ L (Γ0), u := |u| u and (−∆g(t)) is a square root of the Laplace–Beltrami operator on Γ(t).

(4) Another example is the Cahn–Hilliard system on an evolving surface {Γ(t)}t∈[0,T ]

u˙ + u∇g · w = ∆gµ in Γ(t), ′ −∆gu + W (u)= µ, in Γ(t),

u(0) = u0, where W (r) is a given potential. This is analysed in [28] with W (r) = (r2 − 1)2/4, where the authors obtain, for 2 ∞,2 1 2 u0 ∈ H (Γ0), u ∈ W (H ,L ). This has been generalised in [13] by the second and last authors for a wider 1 ∞,2 1 −1 class of potentials and u0 ∈ H (Γ0), where conditions are obtained so that the solution u ∈ W (H ,H ) 2 and µ ∈ LH1 .

7.2 H1(M(t)) pivot space Beside the standard choice of L2 as pivot space, another relevant example for a pivot space is H1. A typical example is the bi-Laplace (also called biharmonic) equation which is a fourth order elliptic PDE and is important in applied mechanics, in particular in the theory of elasticity. The equation is analysed for example in [40, §3, 4.7.5, Example 5]. 1 Let H(t)= H (M(t)) with φt : H0 → H(t) as in (7.2). In this example we work with

X(t)= W 2,r(M(t)) for r ≥ 2.

We start by verifying that φt takes X0 into X(t). We require more regularity for w and Φ, namely

0 3 Rd Rd (·) 1 3 Rd Rd w ∈ C [0,T ]; C ( , ) and Φ0 ∈ C [0,T ]; C ( , ) , (7.8) where d is as in (7.1).

1 2,r Lemma 7.5. Under Assumption 7.1 and (7.8), the pairs (H , φt)t∈[0,T ] and (W , φt)t∈[0,T ] are compatible.

29 We see that for u, v ∈ H0, by using the formula (7.4) for diﬀerentiating the Dirichlet energy,

d ⊺ ⊺ λˆ(t; u, v)= φ uφ v + ∇ φ u ∇ φ v = φ uφ v∇ · w(t)+ ∇ φ u H(t)∇ φ v. dt t t g t g t t t g g t g t ZM(t) ! ZM(t) This then implies for u, v ∈ H(t),

⊺ λ(t; u, v)= uv∇g · w(t)+ ∇gu H(t)∇gv. ZM(t) 1 p q Deﬁnition 7.6 (H (M) weak time derivative). A function u ∈ LX has a weak time derivativeu ˙ ∈ LX∗ if and only if

T T T ⊺ hu˙(t), η(t)iX∗(t),X(t) = − u(t)η ˙(t) − u(t)η(t)∇g · w(t)+ ∇gu(t) H(t)∇gη(t) Z0 Z0 ZM(t) Z0 ZM(t) for all η ∈ DX .

Proposition 7.7. Under Assumption 7.1 and (7.8), for any p, q ∈ [1, ∞], there exists an evolving space equiv- p,q 2,r 2,r ∗ p,q 2,r 2,r ∗ alence between the spaces W (W (M0), W (M0) ) and W (W , (W ) ).

1 Application. We explore an example which motivates the choice of H0 as a pivot space. We present it in the ﬁxed domain setting for simplicity, but it can be easily generalised to an evolving domain or hypersurface. Let Ω ⊂ Rn be a suﬃciently regular bounded domain. We consider the bi-Laplace equation ∂u + ∆2u = f in Ω × (0,T ), (7.9) ∂t ∂∆u u = =0 on ∂Ω × (0,T ), ∂ν u(x, 0) = u0 in Ω.

1 Let H := H0 (Ω) with the standard scalar product (u, v)H := Ω ∇u · ∇v and deﬁne the subspace R n 2 ∂ 2 2 2 ∂ V := v ∈ H : ∆v ∈ L (Ω), i =1,...,n , kvkV := kvkH + ∆v . ∂x ∂x 2 i i=1 i L (Ω) X

The duality pairing between V ∗ and V is deﬁned by

hg, vi ∗ := hg, −∆vi −1 1 . (7.10) V , V H (Ω), H0 (Ω)

2 ∗ We select u0 ∈ H and f ∈ L (0,T ; V ). Taking v ∈ V , we can formally multiply (7.9) by −∆v, integrate by parts and use (7.10) to obtain

′ hu (t), viV ∗, V + ∇(∆u(t)) · ∇(∆v)= hf(t), viV ∗, V , ∀v ∈ V. (7.11) ZΩ By [40, §, Prop. 4.5], there exists u ∈ L2(0,T ; V ) with u′ ∈ L2(0,T ; V ∗) such that (7.11) holds, i.e.,

u′(t) + ∆2u(t)= f(t) in V ∗.

2 1 If we assume more regularity on the forcing term, namely f ∈ L (0,T ; H0 (Ω)), and then set v := −∆g, (7.11) reads as ′ (u (t), v)L2(Ω) + (−∇(∆u), ∇v)L2(Ω) = (f(t), v)L2(Ω). (7.12) So the equation holds weakly for every v in the set W := {v : v = −∆g for some g ∈ V }, which contains H1(Ω), hence (7.12) holds for all v ∈ H1(Ω). By [40, §2, Sect. 9.9], u satisﬁes (7.9).

7.3 H−1(Ω(t)) pivot space The choice of H−1 as a pivot space appears in the study of very weak solutions of certain evolutionary problems following an idea of Brezis [11], see for example [39, §2.3], [49, §III, Example 6.C] and [41]; once we have introduced some notation, we will motivate the study through the porous medium equation. Inspired by this as well as the aforementioned literature, we consider the case of

X(t)= Lp(Ω(t)) ∩ H−1(Ω(t)) and H(t)= H−1(Ω(t)), with p ∈ (1, +∞)

30 Rn −1 1 on a bounded evolving domain {Ω(t)}t∈[0,T ] in , where H (Ω(t)) is the dual space of H0 (Ω(t)) which we endow with the inner product

(u, v) 1 = ∇u · ∇v. H0 (Ω(t)) ZΩ(t) 1 −1 With −∆t : H0 (Ω(t)) → H (Ω(t)) denoting the Dirichlet Laplacian on the domain Ω(t), we endow the pivot space H(t) with the inner product deﬁned by6

−1 (u, v) := hu, (−∆ ) vi −1 1 . H(t) t H (Ω(t)),H0 (Ω(t)) We then identify H(t) ≡ H(t)∗ via the Riesz map (with respect to this inner product). The norm of f ∈ X(t) is deﬁned as

kfkX(t) = kfkLp(Ω(t)) + kfkH−1(Ω(t)),

d and with this, X(t) is a separable and reﬂexive Banach space. Observe that X(t) ֒−→ H(t) as X(t) contains D(Ω(t)). For simplicity of notation, we will denote the Laplacian by

Lt = −∆t.

Remark 7.8. Some important observations are timely: not 1 −1 (i) We do identify H0 (Ω(t)) with H (Ω(t)). (ii) The inner product above indeed deﬁnes a norm on H−1(Ω(t)) which is equivalent to the usual dual norm.

(iii) Since p ∈ (1, ∞) and Lt is uniformly elliptic we have the regularity

p −1 2,p −1 u ∈ L (Ω(t)) =⇒ Lt u ∈ W (Ω(t)) with kLt ukW 2,p(Ω(t)) ≤ CkukLp(Ω(t)) (7.13) by Calder´on–Zygmund theory for elliptic equations (see for instance [34, §9.2]). The constant C > 0 above can be taken to be independent of t.

′ (iv) We identify Lp(Ω(t)) with Lp (Ω(t))∗ so that, in rigour,

′ ′ X(t)= Lp(Ω(t)) ∩ H−1(Ω(t)) ≡ (Lp (Ω(t))∗ ∩ H−1(Ω(t)) and X∗(t)= Lp (Ω(t)) + H−1(Ω(t)).

∗ Given f ∈ X(t) and g = g1 + g2 ∈ X (t), the duality pairing is given by

′ hg,fiX∗(t), X(t) = hg1,fiLp (Ω(t)),Lp(Ω(t)) + (g2,f)H(t) = g1f + (g2,f)H(t). ZΩ(t) ′ This identiﬁcation of Lp(Ω(t)) with Lp (Ω(t))∗ (giving rise to a second identiﬁcation!) does not lead to any contradictions as we do not identify X(t) with X∗(t). In fact, X∗(t) is strictly larger than X(t). (v) If n =1, 2 we have X(t) ≡ Lp(Ω(t)), but in higher dimensions this space is generally strictly smaller than Lp(Ω(t)). Observe however that we have

X(t)= Lp(Ω(t)) if p ≥ 2n/(n + 2),

′ 1 p .as in this case the well-known Sobolev embedding H0 (Ω(t)) ֒→ L (Ω(t)) holds 1 1 As a change of notation, let ψt : H0 (Ω0) → H0 (Ω(t)) be the map that we called φt (defined in (7.2)) in the 0 0 ⊺ 0 previous examples. Note that since we are working over flat domains Ω(t), we have At = (DΦt ) DΦt , which 0 simplifies the formulae (7.5), (7.6), (7.7). In particular, we note that DΦt is invertible and

0 t −1 ψt(DΦt ) = (DΦ0) . We again assume the extra regularity in (7.8) in order to use the results of the previous section. We now deﬁne

∗ φt : H0 → H(t) by φt := (ψ−t) .

6 −1 −1 1 Given u ∈ H (Ω(t)), the function (−∆t) u ∈ H0 (Ω(t)) is the unique weak solution w of the elliptic problem

−∆tw = u on Ω(t), w = 0 on ∂Ω(t).

31 1 The action of this map is as follows: given f ∈ H0, u ∈ H0 (Ω(t)), we have

−1 0 −1 0 −1 hφ f,ui 1 := hf, ψ ui 1 = ∇L f · ∇ψ u = ψ (J ) DΦ ∇ L f · ∇u (7.14) t H(t),H0 (Ω(t)) −t H0,H0 (Ω0) 0 −t t t t g 0 ZΩ0 ZΩ(t) allowing us to identify

0 −1 0 −1 φtf = −∇g · ψt (Jt ) DΦt ∇gL0 f . (7.15)

Analogously, for g ∈ H−1(Ω(t)), we have

0 0 −1 φ−tg = −∇g · Jt DΦt ψ−t(∇g Lt g) .

A A We can perform similar calculations to compute the adjoint maps φt and φ−t: given u ∈ H(t) and v ∈ H0,

0 −1 0 −1 −1 −1 −1 −1 (u, φ v) = (J ) DΦ ∇(ψ L v) · ∇L u = ∇(L v) · ∇(ψ L u)= hv, ψ L ui −1 1 t H(t) t t t 0 t 0 −t t −t t H (Ω0), H0 (Ω0) ZΩ(t) ZΩ0 −1 = (v, L0ψ−tLt u)H0 ,

A A from where we obtain that φt : H(t) → H0 and φ−t : H0 → H(t) satisfy

A −1 A −1 φt u =L0ψ−tLt u and φ−tu =LtψtL0 u. (7.16)

Due to (7.13) these also satisfy

A A φt |X(t) : X(t) → X0 and φ−t|X0 : X0 → X(t).

−1 ∗ It is important to note that since we identify H (Ω(t)) with its dual, the maps φt are also deﬁned with ∗ −1 −1 A φt : H (Ω(t)) → H (Ω0), and up to composition with the Riesz map and its inverse they coincide with φt ∗ ∗ ∗ calculated above. In particular, the map φt = (ψ−t) is not the same as ψ−t. This is another manifestation of 1 the fact that we are not identifying H0 with its dual. p −1 2,p We observe also that, if f ∈ X0, then f ∈ L (Ω0) and due to (7.13) we have L f ∈ W (Ω0). In particular, ′ 0 1 p we can integrate by parts in (7.14) to obtain, for u ∈ H0 (Ω(t)) ∩ L (Ω(t)), the simpler formula

t hφtf,ui 1 = fψ−tu = J ψtfu. (7.17) H(t),H0 (Ω(t)) 0 ZΩ0 ZΩ(t)

−1 p −1 Lemma 7.9. Under Assumption 7.1 and (7.8), the pairs (H , φt)t∈[0,T ] and (L ∩ H , φt)t∈[0,T ] are compatible. The proof of the next lemma is complicated and is given in §7.5. Lemma 7.10. Under Assumption 7.1 and (7.8), we have

−1 −1 λ(t; u, v)= H(t)∇(Lt u) · ∇(Lt v). ZΩ(t) Thus the deﬁnition of a weak time derivative is the following.

−1 p q Deﬁnition 7.11 (H (Ω) weak time derivative). A function u ∈ LX has a weak time derivativeu ˙ ∈ LX∗ if and only if

T T T −1 −1 hu˙(t), η(t)iX∗(t),X(t) = − (u(t), η˙(t))H(t) − H(t)∇(Lt u) · ∇(Lt η) ∀η ∈ DX . Z0 Z0 Z0 ZΩ(t) We can ﬁnally conclude. Proposition 7.12. Under Assumption 7.1 and (7.8), for any p, q ∈ [1, ∞], there exists an evolving space p,q ∗ Wp,q ∗ equivalence between W (X0,X0 ) and (X,X ).

32 Applications. Let us motivate, again in the simpler case of a ﬁxed domain, this choice of pivot space by giving more details for the porous medium equation (PME) as considered in [49, §III, Example 6.C]:

u′ − ∆Ψ(u)= f on (0,T ) × Ω, Ψ(u)=0 on(0,T ) × ∂Ω, (7.18)

u(0) = u0 on Ω,

′ where Ψ(u) := |u|m−1u (or an appropriate generalisation) with m := p − 1 and f ∈ Lp (0,T ; X∗) for X = H−1(Ω) ∩ Lp(Ω). If we take the inner product of the equation in H−1(Ω) with an element g ∈ Lp(Ω) ∩ H−1(Ω), we get ′ −1 (u (t),g)H + Ψ(u(t))g = (−∆) f(t)g ∀g ∈ X. ZΩ ZΩ ′ Suppose that p ≥ 2n/(n + 2) so that X = Lp(Ω). Deﬁne f˜ ∈ Lp (0,T ; X∗) by

hf˜(t), vi := f(t)v for v ∈ X ZΩ and A: X → X∗ and B : X → X∗ by

hA(u), vi := Ψ(u)v and for u, v ∈ H, hBu, vi := (u, v)H , ZΩ ′ it follows by [49, Proposition 6.2, §III.6] that there is a unique u ∈ Lp(0,T ; X) with (Bu)′ ∈ Lp (0,T ; X∗) such that (Bu(t))′ + A(u(t)) = f˜(t) in X∗. ′ ′ Since h(Bu(t)) , vi = (u (t), v)H , this implies that

hu′(t), (−∆)−1gi + hΨ(u(t)),gi = hf˜(t),gi ∀g ∈ X.

Setting v := (−∆)−1g, we get existence of solutions for the very weak formulation of (7.18):

′ 1 p u (t)v + Ψ(u(t))(−∆)v = f(t)(−∆)v ∀v ∈ H0 (Ω) : ∆v ∈ L (Ω). ZΩ ZΩ ′ Under the additional regularity f ∈ Lp (0,T ; H), replacing the deﬁnition of f˜ above by

hf˜(t), vi := v(−∆)−1f(t) for v ∈ X, ZΩ ′ so that f˜ ∈ Lp (0,T ; H1(Ω)), then by [49, Corollary 6.2 and Proposition 6.3, §III.6] we have existence of the ′ 0 ′ p p 1 equation in L (0,T ; H) and Ψ(u) ∈ L (0,T ; H0 (Ω)) (so the boundary condition is satisﬁed). The equation in −1 (7.18) holds pointwise a.e. in time in H (Ω) and the initial condition is satisﬁed in the sense that u(t) → u0 as t → 0 in H−1(Ω). This concept of solution is called as the H−1-solution of the PME. See [54, §6.7] in this context. Of a similar form as this problem is the Stefan problem on a moving domain {Ω(t)}t∈[0,T ]

e˙ − ∆gu + e∇g · w = f in Ω(t),

e(0) = e0, e ∈E(u), where the maximal monotone graph E is deﬁned via

r for r< 0 E(r)= [0, 1] for r =0 , r +1 for r> 0

 1 1 which was considered by the ﬁrst and ﬁnal authors in [1]. For f ∈ LL1 and e0 ∈ L (Ω0), the authors look for 1 ∞ ∞ 2 ∞ u,e ∈ LL1 , and for f ∈ LL∞ and e0 ∈ L (Ω0) one looks for u ∈ LH1 and e ∈ LL∞ ;

33 7.4 Non-Gelfand triple examples In the previous examples we obtained the definition of the weak derivative for three different cases in which there is a pivot Hilbert space, whose inner product structure we could exploit to establish the evolving space equivalence property of the evolving Sobolev–Bochner spaces. To conclude this section, we now consider several examples in which we do not assume the existence of a pivot space. The evolving space equivalence property seems difficult to establish in some of the cases below, so we simply present the definition of the weak time derivative and leave the remaining issues for future consideration. We fix, for all the examples below, r ∈ (1, 2).

Example 1. The simplest example one can consider is obtained by taking X(t) = Lr(Γ(t)) and Y (t) = L1(Γ(t)), where the evolution of {Γ(t)} is determined by the ﬂow map (7.2). As in Remark 4.7, we have p q Πt = IdX0 for all t, and, given p, q ∈ [1, +∞], a function u ∈ LLr has weak time derivativeu ˙ ∈ LL1 if

T T uη˙ = − uη,˙ ∀η ∈ DL∞ . Z0 ZΓ(t) Z0 ZΓ(t) The evolving space equivalence property for Wp,q(Lr,L1) follows immediately. The same is true if we take X(t)= W k,r(Γ(t)), Y (t)= W k,1(Γ(t)), for general k ∈ N.

Example 2. Consider X(t)= W 1,r(Γ(t)) and Y (t)= L1(Γ(t)) where the flow maps of each are defined as in (7.2) but are different to each other, say

X t Y t φt u = u ◦ Φ0 and φt = u ◦ Φ0,

0 0 where Φt and Φt are flows determined by given velocity fields w and w,e respectively. We assume that these have the same normal component (indeed they must otherwise the surfaces will be different) but with potentially different tangentiale parts, say wτ and wτ . In general, we denote quantitiese of interest (such as the determinant t of the Jacobian) using the notation (·) for the corresponding quantity derived from Φ0. It is clear that both X Y Y pairs (X(t), φt )t and (Y (t), φt )t are compatiblee and it is easy to check that the dual maps of φt and its inverse are given by f e

Y ∗ ∞ ∞ Y ∗ t Y (φ−t) : L (Γ0) → L (Γ(t)), (φ−t) v = J0φt v, Y ∗ ∞ ∞ Y ∗ 0 Y (φ ) : L (Γ(t)) → L (Γ0), (φ ) v = J φ v. t t et −t Recall that e

π(t; u, v)= hΠ u, vi 1 ∞ = Π uv, t L (Γ0),L (Γ0) t ZΓ0 Y X and since Πt = φ−tφt , we have by the chain rule

d Y X Y X Y t Y t Y φ φ u = φ φ ∇gu · φ (∂tΦ )+ φ (DΦ )φ w(t) dt −t t −t t −t 0 −t 0 −t (7.19) = φY φX ∇ u · φ Y ∂ Φt + DΦt w(t) −t t g −t t 0 0 e from where we identify e d λˆ(t; u, v)= π(t; u, v)= φY φX ∇ u · φY ∂ Φt + DΦt w(t) v. dt −t t g −t t 0 0 ZΓ0 Pushing forward then yields, for u ∈ X(t) and v ∈ Y ∗(t), e

ˆ X Y ∗ Y X X Y t t Y ∗ λ(t; u, v)= λ(t; φ−tu, (φt ) v)= φ−tφt ∇g(φ−tu) · φ−t ∂tΦ0 + DΦ0 w(t) (φt ) v ZΓ0 Y X X Y t t 0 Y = φ−tφt ∇g(φ−tu) · φ−t ∂tΦ0 + DΦ0 we (t) Jt φ−tv ZΓ0 X X t t e = φt ∇g(φ−tu) · ∂tΦ0 + DΦ0 w(t) ve ZΓ(t) X 0 ⊺ t t = φt (DΦt ) ∇gu · ∂tΦ0 + ∇geu · DΦ0 w(t) v ZΓ(t) e 34 X 0 ⊺ t t ⊺ = φt (DΦt ) ∇gu · ∂tΦ0 + (DΦ0) ∇gu · w(t) v ZΓ(t) X 0 ⊺ t t ⊺ = φt (DΦt ) ∇gu · ∂tΦ0 + (DΦ0) ∇gu · we τ (t) v ZΓ(t) X 0 ⊺ t t = φt (DΦt ) ∇gu · ∂tΦ0 + ∇gu · DΦ0wτ (et) v ZΓ(t) X 0 ⊺ t t = φt (DΦt ) ∇gu · ∂tΦ0 + DΦ0wτ (t) ev. ZΓ(t) (·) 1 2 Rd Rd e Let us assume that Φ0 ∈ C ([0,T ]; L ( ; )). Diﬀerentiating with respect to t the identity

t 0 Φ0 ◦ Φt (p)= p, p ∈ Γ0, we obtain, for all p ∈ Γ0,

t 0 t 0 0 t 0 t 0 0 0 = (∂tΦ0)(Φt (p)) + DΦ0(Φt (p))∂tΦt (p) = (∂tΦ0)(Φt (p)) + DΦ0(Φt (p))w(t, Φt (p)).

Pushing forward to Γ(t) the above is equivalent to

t t ∂tΦ0 + DΦ0 w =0.

Plugging this in the expression above, we ﬁnd

X 0 ⊺ t t λ(t; u, v)= φt (DΦt ) ∇gu · DΦ0wτ (t) − DΦ0w(t) v ZΓ(t) X 0 ⊺ t = φt (DΦt ) ∇gu · DΦ0 (we τ (t) − wτ (t)) v. ZΓ(t) p e q We then conclude that, given p, q ∈ [1, +∞], a function u ∈ LW 1,r has weak time derivativeu ˙ ∈ LL1 if

T T T ⊺ X 0 t ∞ uη˙ = − uη˙ − φt (DΦt ) ∇gu · DΦ0 (wτ (t) − wτ (t)) η ∀η ∈ DL . Z0 ZΓ(t) Z0 ZΓ(t) Z0 ZΓ(t) Remark 7.13. Observe that in the case where w and w have the samee tangential component, we do indeed recover the situation of Example 1. e 2,r 1,1 Example 3. As a ﬁnal example we take X(t)= W0 (Ω(t)) and Y (t)= W0 (Ω(t)), under the same assumptions as the previous case. In this case, Πt has the same formula as above, but we note that

π(t; u, v)= hΠtu, vi 1,1 −1,∞ W0 (Ω0), W (Ω0)

−1,∞ −1,∞ and so we need a representation for elements of W (Ω0). By [12, Proposition 9.20], given F ∈ W (Ω0), ∞ there exist f1,...,fn ∈ L (Ω0) such that

hF,ui −1,∞ 1,1 = fiDiu, W (Ω0), W0 (Ω0) i=1 Ω0 X Z so writing f = (f1,...,fn) we can identify F with the operator −∇ · f. We also need to identify the adjoint of Y ∗ φt : given u ∈ Y0 and v ∈ Y (t), we have

Y Y t ⊺ Y v, φt u −1,∞ 1,1 = v · ∇(φt u)= v · (DΦ0) φt (∇u) W (Ω(t)),W0 (Ω(t)) ZΩ(t) ZΩ(t)

t e Y = DΦ0v · φt (∇u) ZΩ(t) 0 e 0 −1 Y = Jt (DΦt ) φ−tv · ∇u ZΩ0 from where we identify e e

Y ∗ −1,∞ −1,∞ Y ∗ 0 0 −1 Y (φt ) : W (Ω(t)) → W (Ω0), (φt ) v = −∇ · Jt (DΦt ) φ−tv . e e 35 We then have

π(t; u, v)= v · ∇Πtu, ZΩ0 and recalling the formula in (7.19) this leads to d d λˆ(t; u, v)= π(t; u, v)= v · ∇ Π u = v · ∇ φY φX ∇ u · φY ∂ Φt + DΦt w(t) . dt dt t −t t g −t t 0 0 ZΩ0 ZΩ0 1,r −1,∞ We ﬁnally push forward to time t: given u ∈ W0 (Ω(t)), v ∈ W (Ω(t)), e

X Y ∗ 0 0 −1 Y Y X X Y t t λ(t; u, v)= λ(t; φ−tu, (φt ) v)= Jt (DΦt ) φ−tv · ∇ φ−tφt ∇g(φ−tu) · φ−t ∂tΦ0 + DΦ0w(t) ZΩ0 b e0 e0 −1 Y Y t −⊺ Y t t = Jt (DΦt ) φ−tv · ∇ φ−t((DΦ0) ∇u · φ−t ∂tΦ0 + DΦ0we (t) ZΩ0 eY e t −1 t Y t −⊺ t −⊺ t t = φ−t (J0) (DΦ0)v · φ−t (DΦ0) ∇ (DΦ0) ∇u · ∂tΦe0 + DΦ0w(t) ZΩ0 h i h i t e et −⊺ t −e⊺ t t = (DΦ0)v · (DΦ0) ∇ (DΦ0) ∇u · ∂tΦ0 + DΦ0w(t) e ZΩ(t) e t e−⊺ t t = v · ∇ (DΦ0) ∇u · ∂tΦ0 + DΦ0w(t) . e ZΩ(t) (·) 1 2 Rd Rd e Again assuming that Φ0 ∈ C ([0,T ]; L ( ; )), we can reason as in the previous example to obtain t t ∂tΦ0 + DΦ0w =0, so that

t −⊺ t λ(t; u, v)= v · ∇ (DΦ0) ∇u · DΦ0 (w(t) − w(t)) . ZΩ(t) p q Hence in this case, given p, q ∈ [1, +∞], a function u ∈ LW 2,r has weake time derivativeu ˙ ∈ LW 1,1 if T T T t −⊺ t uη˙ = − uη˙ − ∇ (DΦ0) ∇u · DΦ0 (w(t) − w(t)) · η, ∀η ∈ DW −1,∞ . Z0 ZΓ(t) Z0 ZΓ(t) Z0 ZΩ(t) e 7.5 Proofs of compatibility and evolving space equivalence We now provide the proofs of the results stated in §7.1–7.3. For readability we restate all the results.

7.5.1 L2 pivot space

2 1,r Lemma 7.2. Under Assumption 7.1, the pairs (L , φt)t∈[0,T ] and (W , φt)t∈[0,T ] are compatible. r Proof. If v ∈ L (M0), we have

r r t r r |φtv| = |v| J0 ≤ CJ kvkL (M0), ZM(t) ZM0 r r t which shows that φt : L (M0) → L (M(t)) is bounded. Moreover, since t 7→ J0 is continuous, so is t 7→ 2 2 kφtvkL2(M(t)) and hence (L (M(t)), φt)t is indeed compatible. Using (7.5), the remaining claim follows from

r 0 0 −1 r 0 r |∇ φ u| = |D Φ (A ) ∇ u| J ≤ C k∇ uk p . g(t) t g0 t t g0 t g L (M0) ZM(t) ZM0 Proposition 7.4. Under Assumption 7.1, given r ≥ 2 and for any p, q ∈ [1, ∞], there exists an evolving space p,q 1,r 1,r ∗ p,q 1,r 1,r ∗ equivalence between the spaces W (W (M0), W (M0) ) and W (W , (W ) ). 0 −1 0 Proof. From §7.1, we see that Πt : H0 → H0 is deﬁned by Πtu := uJt with inverse Πt u := u/Jt . The regularity 0 0 −1 1 1 Rd Rd assumptions about the velocity ﬁeld imply that J(·), (J(·)) ∈ C ([0,T ], C ( , )) and hence

1,r 1,r kΠtukW (M0) ≤ CkukW (M0)

∞ 1,∞ 0 −1 where C depends on the L (0,T ; W (M0)) norm of Jt . We can prove in the same way that the inverse Πt −1 p,q p,q is bounded as well. It is not diﬃcult to check that Π : W (X0,X0) →W (X0,X0) due to the smoothness 0 assumptions on Φt and hence the evolving space equivalence holds by Theorem 5.6.

36 7.5.2 H1 pivot space

1 2,r Lemma 7.5. Under Assumption 7.1 and (7.8), the pairs (H , φt)t∈[0,T ] and (W , φt)t∈[0,T ] are compatible.

1 1,r Proof. The fact that (H , φt)t∈[0,T ] is compatible was proved in Lemma 7.2; the extension to W for r> 2 is 2,r 2,r 2,r trivial. We now show that φt : W (M0) → W (M(t)) is bounded uniformly. If v ∈ W (M0), we already 1,r 1,r know that φtv ∈ W (M(t)) and it suﬃces to prove that ∂i(φtv) ∈ W (M(t)) for each i. From (7.5), we have

0 0 −1 ∂i(φtv)= φt((DΦt )ik(At )kj ∂j v), Xj,k 0 0 −1 1,r 1,r and hence we need to show that (DΦt )ik(At )kj ∂j v ∈ W (M0) (since we know that φt preserves W 0 1,r regularity) for all i, j, k. This is true due to the regularity assumptions on Φ(·) and because ∂j v ∈ W (M0). The uniform boundedness of φt : X0 → X(t) also follows. We can prove in the same way that the inverse map φ−t satisﬁes a similar estimate. A lengthy calculation as in the proof of Lemma 7.2 shows that t 7→ k∂ij (φtv)kLp(M(t)) is continuous, and this ﬁnishes the proof.

p,q 2,r 2,r ∗ The proof of Proposition 7.7 (evolving space equivalence between the spaces W (W (M0), W (M0) ) and Wp,q(W 2,r, (W 2,r)∗)) requires us to check the conditions of Theorem 5.6, which we will do now in a series of lemmas.

Firstly, writing (Πtu, v)H0 = (φtu, φtv)H(t) and at the same time expanding the inner product on the left- hand side,

0 0 0 −1 ⊺ 0 0 −1 0 (Πtu, v)H0 = Πtuv + ∇gΠtu · ∇gv = uvJt + (DΦt (At ) ∇gu) DΦt (At ) ∇gvJt ZM0 ZM0 0 ⊺ 0 = uvJt + ∇gu Bt ∇gv, ZM0 where we denoted 0 0 −⊺ 0 ⊺ 0 0 −1 0 Bt := (At ) (DΦt ) DΦt (At ) Jt . (7.20)

By comparing these two expressions, we are able to obtain relevant properties of Πt.

Lemma 7.14. Under Assumption 7.1 and (7.8), we have Πt : X0 → X0 is a bounded linear map.

Proof. Given u ∈ X0, setting w = Πtu, we have by the above displayed equation

⊺ 0 ⊺ 0 wv + ∇gw ∇gv = uvJt + ∇gu Bt ∇gv ∀v ∈ H0. (7.21) ZM0 ZM0 1 ∗ As a function of v, the right-hand side is clearly an element of H (M0) , so by the Lax–Milgram lemma, there 1 exists a unique w ∈ H (M0) satisfying the above equation. By smoothness, we can rewrite this as

⊺ 0 0 wv + ∇gw ∇gv = (uJt − ∇g · (Bt ∇gu))v ∀v ∈ H0, ZM0 ZM0 0 0 p i.e., w is a weak solution w−∆Γw = (uJt −∇g ·(Bt ∇gu)) ∈ L (Γ0). We may apply elliptic regularity theory (by making use of the usual estimates, e.g. [34, §9.2] on Euclidean balls and using a patching argument to extend to 2,r the manifold case if M(t) =Γ(t), as is standard) to this variational formulation to deduce that w ∈ W (Γ0) as well as 0 0 kwk 2,r ≤ C uJ − ∇g · (B ∇gu) r . W (Γ0) t t L (Γ0)

Lemma 7.15. Under Assumption 7.1 and (7.8) , the map Πt : X0 → X0 is invertible with uniformly bounded −1 −1 p p inverse with t 7→ Πt w measurable. Hence Π : L (0,T ; X0) → L (0,T ; X0).

Proof. In this case, one needs to show that given w ∈ X0, there exists u ∈ X0 such that (7.21) holds and the proof is almost identical to the previous lemma after realising that the right-hand side of (7.21) is an equivalent 0 0 inner product on H0. The measurability follows because Jt and Bt are continuous. −1 p,q p, p∧q Lemma 7.16. Under Assumption 7.1 and (7.8), we have Π : W (X0,X0) → W (X0,X0) for any p, q ∈ [1, ∞].

37 −1 1 Proof. We shall ﬁrst show that Π : C ([0,T ]; X0) → W(X0,X0) and then extend by density. Take w ∈ 1 −1 C ([0,T ]; X0) and set u = Π w. We have that u(t) satisﬁes

0 ⊺ 0 ⊺ u(t)vJt + ∇gu(t) Bt ∇gv = w(t)v + ∇gw(t) ∇gv ∀v ∈ H0. ZM0 ZM0

Taking the diﬀerence at times t + h and t, this becomes, for all v ∈ H0,

0 0 ⊺ 0 ⊺ 0 ⊺ δhu(t)vJt+h + u(t)δhJt v + ∇gδhu(t) Bt+h∇gv + ∇gu(t) δhBt ∇gv = δhw(t)v + ∇gδhw(t) ∇gv. ZM0 ZM0 Now, adding and subtracting y(t) where y(t) is deﬁned as the solution of

0 0 ′ ′ 0 ′ 0 ′ y(t)vJt + ∇gy(t)Bt ∇gv = w (t)v + ∇gw (t)∇gv − u(t)v(Jt ) − ∇gu(t)(Bt ) ∇gv ∀v ∈ H0, (7.22) ZM0 ZM0 we obtain

0 0 ⊺ 0 ⊺ 0 (δhu(t) − y(t)) vJt+h + u(t)δhJt v + ∇g δhu(t) − y(t) Bt+h∇gv + ∇gu(t) δhBt ∇gv ZM0 0 0 ⊺ + y(t)vJt+h + ∇gy(t)Bt+h∇gv = δhw(t)v + ∇gδhw(t) ∇gv. ZM0 ZM0 Observe that, using the deﬁnition of y(t), the ﬁnal term on the left-hand side is

0 0 0 0 0 0 y(t)vJt+h + ∇gy(t)Bt+h∇gv = y(t)v(Jt+h − Jt )+ ∇gy(t)(Bt+h − Bt )∇gv ZM0 ZM0 ′ ′ 0 ′ 0 ′ + w (t)v + ∇gw (t)∇gv − u(t)(Jt ) v − ∇gu(t)(Bt ) ∇gv, ZM0 so the above becomes

0 0 0 ′ ⊺ 0 (δhu(t) − y(t)) vJt+h + u(t) δhJt − (Jt ) v + ∇g δhu(t) − y(t) Bt+h∇gv ZM0 ⊺ 0 0 ′ 0 0 0 0 + ∇gu(t) δhBt − (Bt ) ∇gv + y(t)v(Jt+h − Jt )+ ∇gy(t)(Bt+h − Bt )∇gv ZM0 ZM0 ′ ⊺ ′ = (δhw(t) − w (t)) v + ∇g δhw(t) − ∇gw (t) ∇gv. ZM0 Taking v = δhu(t) − g and using Young’s inequality with ǫ multiple times, we ﬁnd

2 ′ 2 0 0 ′ 2 2 C kδhu(t) − y(t)k ≤kδhw(t) − w (t)k + δhJ − (J ) ∞ ku(t)k H0 H0 t t L (M0) 0 0 ′ 2 2 0 0 2 2 + δhB − (B ) ∞ k∇gu(t)k 2 + J − J ∞ ky(t)k 2 t t L (M 0) L (M0) t+h t L (M0) L (M0) 0 0 2 2 + B − B ∞ k∇gy(t)k 2 , t+h t L (M0) L (Ω)

−1 1 ′ which shows that u is strongly diﬀerentiable; more precisely, u = Π w ∈ C ([0,T ]; H0) with u = g. By the same reasoning as the previous lemma applied to the weak formulation for y(t) (see (7.22)), we obtain in fact that −1 ′ ′ −1 (Π w(t)) 2,r ≤ C(kw (t)k 2,r + Π w(t) 2,r ). t W (M0) W (M0) t W (M0) −1 1 p, p∧q −1 p,q p, p∧q Hence Π : C ([0,T ]; X0) →W (X0,X0) is such that Π : W (X0,X0) →W (X0,X0) is bounded. By density then, we obtain the result. Proposition 7.7. Under Assumption 7.1 and (7.8), for any p, q ∈ [1, ∞], there exists an evolving space equiv- p,q 2,r 2,r ∗ p,q 2,r 2,r ∗ alence between the spaces W (W (M0), W (M0) ) and W (W , (W ) ). Proof. Having checked all conditions of Theorem 5.6 above, the result follows.

38 7.5.3 H−1 pivot space

−1 p −1 Lemma 7.9. Under Assumption 7.1 and (7.8), the pairs (H , φt)t∈[0,T ] and (L ∩ H , φt)t∈[0,T ] are compatible.

−1 Proof. The pair (H , φt)t∈[0,T ] is compatible as argued in §3.1. Combining (7.13) with (7.15), it also follows that φt|X0 : X0 → X(t) is bounded and invertible, with inverse φ−t|X(t) satisfying the same properties. Finally, we observe that, given f ∈ X0, we have

kφtfkX(t) = kφtfkLp(Ω(t)) + kφtfkH−1(Ω(t)).

−1 Now the second term is measurable because (H , φt)t∈[0,T ] is compatible, and for the ﬁrst term we use (7.7) p t −1 together with the fact that f ∈ L (Ω0) to write φtf = J0 ψt(∆L0 f), from where

p t p −1 p 0 1−p −1 p kφtfkLp = (J0) ψt(∆L0 f) = (Jt ) |∆L0 f| , ZΩ(t) ZΩ0 which is a continuous function of t, hence also measurable. This proves compatibility of the second pair. To provide the expression for λ in Lemma 7.10, we now verify that Assumption 4.5 is satisfied. Given 2 1 u ∈ H0, we must check that kφtukH(t) is differentiable. Define w(t) ∈ H0 (Ω(t)) by

Ltw(t)= φtu, (7.23) so that, as we argued above, 2 2 kφtukH(t) = |∇tw(t)| . (7.24) ZΩ(t) ∞ Observe that the right-hand side of (7.23) is clearly in CH with zero time derivative, and hence as is the ∞ left-hand side, i.e., Lw ∈ CH with ∂•(Lw)=0. 1 To prove that (7.24) is diﬀerentiable, we need w itself to belong to C 1 , which the next lemma shows is the case. H0 0 1 ∞ We shall henceforth assume that A(·) ∈ C ([0,T ]; L (Ω0)). In the proof below we make use of the notation δh again to denote the diﬀerence quotient.

−1 1 Lemma 7.17. Under Assumption 7.1 and (7.8), for u ∈ H0, we have w ≡ L φ(·)u ∈ C 1 and w˙ satisﬁes, for (·) H0 all t ∈ [0,T ], ⊺ 1 ∇w˙ (t) · ∇ϕ = − ∇w(t) H(t)∇ϕ ∀ϕ ∈ H0 (Ω(t)). ZΩ(t) ZΩ(t) 1 1 1 Proof. Let us show that w ∈ C 1 by proving thatw ˜ := ψ−(·)w ∈ C ((0,T ); H0 (Ω0)). Due to (7.5) and reusing H0 0 the notation Bt from (7.20), we see from (7.23) thatw ˜ satisﬁes

⊺ 0 ∗ 1 ∇w˜(t) Bt ∇ψ−tϕ = hψ−tu, ϕi = hu, ψ−tϕi ∀ϕ ∈ H0 (Ω(t)). ZΩ0 Hence

⊺ 0 1 ∇w˜(t) Bt ∇η = hu, ηi ∀η ∈ H0 (Ω0). (7.25) ZΩ0 Take two times t,s ≥ 0 and consider the diﬀerence of the above equality at those times:

⊺ ⊺ 0 ⊺ 0 0 ∇(˜w(t) − w˜(s) )Bt ∇η +w ˜(s) (Bt − Bs)∇η =0. ZΩ0 Taking η =w ˜(t) − w˜(s), this implies the bound

0 0 C k∇w˜(t) − ∇w˜(s)k 2 ≤kw˜(s)k 2 B − A ∞ , L (Ω0) L (Ω0) t s L (Ω0)

0 1 and the right-hand side clearly tends to zero as t → s, proving that w ˜ ∈ C ([0 ,T ]; H0 (Ω0)). Regarding the derivative, let h > 0 and take the diﬀerence in (7.25) between times t + h and t and divide by h: 0 ⊺ 0 ∇δhw˜(t)Bt+h∇η + ∇w˜(t) δhBt ∇η =0. (7.26) ZΩ0 ZΩ0

39 1 We now show that the diﬀerence quotient forw ˜(t) converges to the (unique) solution v(t) ∈ H0 (Ω(t)) of

⊺ 0 ⊺ 0 ′ 1 ∇v(t) Bt ∇η = − ∇w˜(t) (Bt ) ∇η ∀η ∈ H0 (Ω0). ZΩ0 ZΩ0 In (7.26), if we add and subtract the same term, we see

⊺ 0 ⊺ 0 ⊺ 0 ∇(δhw˜(t) − v(t)) Bt+h∇η + ∇v(t) Bt+h∇η + ∇w˜(t) δhBt ∇η =0, ZΩ0 and here adding and subtracting ∇v(t)⊺B0∇η and using the equation deﬁning v(t), we end up with Ω0 t R ⊺ 0 ⊺ 0 0 ⊺ 0 0 ′ (∇δhw˜(t) − ∇v(t)) Bt+h∇η + ∇v(t) Bt+h − Bt ∇η + ∇w˜(t) δhBt − (Bt ) ∇η =0. ZΩ0 Taking η appropriately, using positive-deﬁniteness and smoothness of A, we get

0 0 0 0 ′ C kδh∇w˜(t) − ∇v(t)k 2 ≤ k∇v(t)k 2 B − B ∞ + k∇w˜(t)k 2 δhB − (B ) ∞ , L (Ω0) L (Ω0) t+h t L (Ω0) L (Ω0) t t L (Ω0) and in the limit h → 0, the right-hand side tends to zero and hence

1 δhw˜(t) → v(t) in H0 (Ω0) but then we must have thatw ˜′ exists andw ˜′ ≡ v. By considering the equation deﬁning v =w ˜′ and making a ′ 0 1 similar argument to how we showed thatw ˜ is continuous, we can show thatw ˜ ∈ C ([0,T ]; H0 (Ω0)). Pushing forward the integrals deﬁningw ˜′(t), we see that

⊺ 0 ′ t ⊺ 0 0 ′ 0 ⊺ ∇w˜(t) (Bt ) ∇η = J0∇w(t) ψt(DΦt )ψt((Bt ) )ψt(DΦt ) ∇ϕ ZΩ0 ZΩ(t) t ⊺ t −1 0 ′ t −⊺ = J0∇w(t) (DΦ0) ψt((Bt ) )(DΦ0) ∇ϕ. ZΩ(t) The identity in Lemma A.2 gives a simpliﬁcation of the right-hand side above and provides the desired result. Lemma 7.10. Under Assumption 7.1 and (7.8), we have

−1 −1 λ(t; u, v)= H(t)∇(Lt u) · ∇(Lt v). ZΩ(t) Proof. Using the transport formula (7.4) on (7.24) and plugging the result of the previous lemma in, we derive

d kφ uk2 = 2∇w˙ (t) · ∇w(t) − H(t)∇w(t) · ∇w(t)= H(t)∇w(t) · ∇w(t). dt t H(t) ZΩ(t) ZΩ(t) We have then that 1 d d λˆ(t; u , v ) := kφ (u + v )k2 − kφ (u − v )k2 0 0 4 dt t 0 0 H(t) dt t 0 0 H(t) 1 = H(t)∇z(t) · ∇z(t) − H(t)∇y(t) · ∇y(t) , 4 Ω(t) Ω(t) Z Z ∗ ∗ where z(t) and y(t) are deﬁned via Ltz(t)= ψ−t(u0 + v0) and Lty(t)= ψ−t(u0 − v0). Deﬁning also

∗ ∗ Ltw(t)= ψ−tu0 and Ltv(t)= ψ−tv0, and using linearity, the above simpliﬁes to

ˆ −1 ∗ −1 ∗ λ(t; u0, v0)= H(t)∇w(t) · ∇v(t)= H(t)∇(Lt ψ−tu0) · ∇(Lt ψ−tv0), ZΩ(t) ZΩ(t) and a simple calculation shows that Assumptions 4.5 (ii), (iii) (see Remark 4.13) are also satisﬁed. Pushing forward to Ω(t) now yields the desired expression. We now check the evolving space equivalence result for this example again by verifying the assumptions of Theorem 5.6.

40 Lemma 7.18. Under Assumption 7.1 and (7.8), for u ∈ H0, we have

−1 Πtu =L0ψ−tLt φtu and Πt : X0 → X0 is uniformly bounded and invertible with uniformly bounded and measurable (in time) inverse.

Proof. The formula follows directly from (7.16). Recalling that φt (resp. φ−t) maps X0 to X(t) (resp. X(t) to X0) and is bounded, we can easily see that Πt : X0 → X0 is bounded due to the elliptic regularity of (7.13). It −1 −1 −1 also has an inverse defined by Πt = φ−tLtψtL0 with the same properties. Measurability of Πtu, Πt u, for u ∈ H0, follows from the fact that the composition of measurable maps is measurable. Thus, we have the fulfillment of (5.5), (5.6) and (5.7). Lemma 7.19. Under Assumption 7.1 and (7.8), the conditions of Theorem 5.6 are fulfilled. Proof. We shall make use of the alternative criteria provided in Lemma 5.8 here to verify (5.8). First, as we 1 already stated, note that H0 (Ω(t)), ψt t is a compatible pair and furthermore, Assumption 4.5 is satisfied (the ψ ˆψ 7 associated operators π , λ satisfy the conditions in Remark 4.13). In this setting, we have (see Definition 7.6)

ψ λˆ (t; u, v)= H(t)∇(ψtu) · ∇(ψtv). (7.27) ZΩ(t) (5.13): Deﬁning H A ξt := (φ−t) , it follows from separability of H0 that (H(t), ξt)t is a compatible pair. We now observe that, for ﬁxed u ∈ H0,

2 −1 2 −1 2 kξtuk = kLtψtL uk = kψtL uk 1 , H(t) 0 H(t) 0 H0 (Ω(t))

1 and this is continuously diﬀerentiable since H0 (Ω(t)), ψt t∈[0,T ] satisﬁes Assumption 4.5. The remaining points follow immediately; simply note that from the calculation above we obtain ξ −1 −1 ψ −1 −1 ξ ψ −1 −1 π (t; u, v) = (ψ L u, ψ L v) 1 = π (t;L u, L v) =⇒ λˆ (t; u, v)= λˆ (t;L u, L v). t 0 t 0 H0 (Ω(t)) 0 0 0 0

ξ ψ (5.14): If u ∈ X0 and v ∈ H0, then we have from using the relation between λˆ and λˆ and the formula for the latter in (7.27) that

ˆξ −1 −1 λ (t; u, v)= − ∇ · (H(t)∇(ψtL0 u)), ψtL0 v −1 1 H (Ω(t)), H0 (Ω(t)) −1 t −1 = − L0 J0ψ−t∇ · (H(t)∇(ψtL0 u)) , v −1 1 H (Ω0), H0 (Ω0) where we used (7.17) (or rather the inverse of the expression given by that formula) since u ∈ X0 and the fact −1 that L0 is self-adjoint in the above manipulation. With this, we can identify ˆ ξ −1 t −1 Λ (t)u = −L0 J0ψ−t∇ · (H(t)∇(ψtL0 u))

p ξ p and as u ∈ L (Ω0) then (7.13) implies that also Λˆ (t)u ∈ L (Ω0), proving the claim.

−1 −1 (5.15): We have already shown that Πt : X0 → X0 is a bijection with inverse given by Πt u = φ−tLtψtL0 u, and from this formula we can immediately identify

−1 ∗ ∗ ∗ −1 ∗ −1 ∗ (Πt ) : X0 → X0 , (Πt ) f =L0 φ−tLtφ−tf.

In particular, if f ∈ X0, we have

−1 ∗ −1 ∗ −1 A (Πt ) f =L0 φ−tLtφ−tf =L0 φ−tLtφ−tf, which is bounded due to (7.13) and the formula in (7.16). Now an application of Theorem 5.6 yields the following. Proposition 7.12. Under Assumption 7.1 and (7.8), for any p, q ∈ [1, ∞], there exists an evolving space p,q ∗ Wp,q ∗ equivalence between W (X0,X0 ) and (X,X ). 7 1 The inner product we use on H0 (Ω(t)) has no lower order term and thus no term involving the divergence of the velocity appears in the expression deﬁning λˆψ.

41 8 Well-posedness for a nonlinear monotone equation

In this ﬁnal section, we establish some results regarding existence and uniqueness of weak solutions for a class of nonlinear equations in order to illustrate the applicability of the functional framework developed in the text and how it can be used to formulate general problems in a fully Banach space setting. Let p ∈ (1, ∞). For generality, we consider a family of (not necessarily linear) operators A(t): X(t) → X∗(t) deﬁned on a separable Banach space X(t) satisfying the following properties: for all u, v ∈ X(t),

(i) (Measurability) the map t 7→ hA(t)u, viX∗(t),X(t) is measurable

(ii) (Monotonicity) hA(t)u − A(t)v,u − viX∗(t), X(t) ≥ 0

(iii) (Hemicontinuity) the map s 7→ hA(t)(u + sv), viX∗(t),X(t) is continuous (from R to R) ′ p (iv) (Boundedness) there exists a constant Cb > 0 independent of t and cb ∈ L (0,T ) such that

p−1 hA(t)u, viX∗(t), X(t) ≤ CbkukX(t)kvkX(t) + cb(t)kvkX(t)

(v) (Coercivity) there exist C c > 0 and cc ≥ 0 independent of t such that

p hA(t)u,uiX∗(t), X(t) ≥ CckukX(t) − cc.

These are the standard assumptions that are made for nonlinear monotone problems [60, §30.2]. We assume a Gelfand triple structure X(t) ⊂ H(t) ⊂ X∗(t) ∗ and we suppose that X(t) and H(t) are evolving under a map φt and X (t) is evolving under the dual map ∗ φ−t, such that

∗ ∗ (X(t), φt)t∈[0,T ], (H(t), φt)t∈[0,T ], (X (t), φ−t)t∈[0,T ] W ∗ ∗ are all compatible pairs. Furthermore, we assume the equivalence of (X,X ) and W(X0,X0 ). We refer to the previous section for examples of such spaces and proofs of the evolving space equivalence. Deﬁning the superposition operator (Au)(t)= A(t)u(t), we consider the equation

′ p u˙ + Au +Λu = f in L ∗ , X (8.1) u(0) = u0 in H0.

′ ′ p Wp,p ∗ Deﬁnition 8.1. Given f ∈ LX∗ and u0 ∈ H0, a weak solution of (8.1) is a function u ∈ (X,X ) satisfying

T T p ∗ ∗ ∗ hu˙(t), v(t)iX (t), X(t) + hA(t)u(t), v(t)iX (t), X(t) + λ(t; u(t), v(t)) = hf(t), v(t)iX (t), X(t) ∀v ∈ LX , Z0 Z0 u(0) = u0.

Our aim in this section is to prove the next result.

′ p Theorem 8.2. Under the above assumptions, given f ∈ L ∗ and u0 ∈ H0, there exists a unique weak solution ′ X u ∈ Wp,p (X,X∗) to (8.1). A concrete example of (8.1) is the evolutionary p-Laplace equation as the next example demonstrates. Aside from this, equations of the form (8.1) may arise as regularisations of PDEs with a more complicated structure. Example 8.3 (The p-Laplace equation on an evolving surface/domain). Let p ∈ (1, ∞) and take M(t) to be an evolving surface Γ(t) ⊂ R3 or domain Ω(t) ⊂ R2. Deﬁne

1,p W0 (M(t)) : if M(t)=Ω(t) X(t)= 1,p (W (M(t)) : if M(t)=Γ(t)

p ∗ and the p-Laplace operator −∆g(t) : X(t) → X (t) which has the action

p ∗ p−2 h−∆g(t)u, viX (t),X(t) := |∇g(t)u| ∇g(t)u · ∇g(t)v. ZM(t)

42 Take a constant α> 08. We consider the equation9

p p−2 u˙ − ∆gu + αu|u| + u∇g · w = f,

u(0) = u0, which, if M(t)=Ω(t), we supplement with the boundary condition u =0 on ∂Ω(t). The operator A is deﬁned p−2 p by A(t)(u) := αu|u| − ∆g(t)u. We have the Gelfand triple structure

X(t) ⊂ L2(M(t)) ⊂ X∗(t).

This is obvious if p ≥ 2, and in the case 1 ≤ p< 2, recalling that the dimension of the manifold is 2, it follows from the Sobolev embedding

.((W 1,p(M(t)) ֒→ L2p/(2−p)(M(t)) ֒→ L2(M(t

Evidently, the pivot space is H(t) = L2(M(t)) and we are in the setting of §7.1 from where we identify the extra term in the deﬁnition of the time derivative to be Λ(t)u = u∇g · w and we also have the evolving space equivalence property.

8.1 Proof of well-posedness We now proceed to establish existence, uniqueness and stability of weak solutions via the Faedo–Galerkin 0 t method. We start by choosing an orthogonal basis {wj }j∈N for X0 and transport it along the ﬂow to {wj := 0 φtwj }j∈N, which forms a basis for X(t) satisfying the following useful property t N w˙ j ≡ 0 ∀j ∈ .

Deﬁne the approximation spaces

t t p p Vn(t) = span{w1,...,wn} and LVn := {η ∈ LX : η(t) ∈ Vn(t), t ∈ [0,T ]} .

p p t It follows that ∪nLVn is dense in LX . We also make use of the projection operator Pn : H(t) → Vn(t) ⊂ X(t) determined by the formula

t (Pnh − h, ϕ)H(t) = 0 for all ϕ ∈ Vn(t).

Lemma 8.4. N p For each n ∈ , there exists a unique solution un ∈ LVn to the Galerkin approximation

p ∗ ∗ ∗ hu˙ n(t), v(t)iX (t), X(t) + hA(t)un(t), v(t)iX (t), X(t) + λ(t; un(t), v(t)) = hf(t), v(t)iX (t), X(t) ∀v ∈ LVn , 0 (8.2) un(0) = Pn u0,

n n t of the form un(t)= j=1 uj (t)wj . The proof of thisP lemma is standard and is relegated to the appendix.

A priori estimates Test v = un in (8.2) to obtain, using Young’s inequality with ǫ and coercivity of the operator A,

′ 1 d 2 p p p 2 ku (t)k + C ku (t)k ≤ C kf(t)k ∗ + ǫku (t)k + C ku (t)k + c . 2 dt n H(t) c n X(t) 1 X (t) n X(t) 2 n H(t) c

Choosing ǫ = Cc/2 we can manipulate the above and integrate it to get

t ′ t 2 p 2 p 2 kun(t)kH(t) + Cc kunk ≤ku0kH +2C1kfk p′ +2C2 kunkH(t) +2Tcc, X(t) 0 L ∗ Z0 X Z0 whence an application of Gronwall’s inequality implies that

∞ p (un)n is uniformly bounded in LH ∩ LX , (8.3)

8The choice of α = 0 is also possible if M(t) =Ω(t). 9Observe that selecting p = 2 and α = 0 recovers the heat equation.

43 (un(T ))n is uniformly bounded in H(T ).

p Observe also that, due to H¨older’s inequality, we have for all η ∈ LX

T T p−1 hAun, ηi ∗ ≤ Cbkunk kηkX + cb(t)kηkX(t) ≤ Cbkunk p kηk p + kcbk p′ kηk p , X (t),X(t) X LX LX L (0,T ) LX 0 0 Z Z

whence ′ p (Aun)n is uniformly bounded in LX∗ . (8.4)

t Remark 8.5. It is an open question whether Pn : Vn(t) → Vn(t) is bounded uniformly in n when t> 0. Without p an aﬃrmative answer, it becomes more challenging to obtain a bound on u˙ n in the dual space LX∗ by the usual duality method but it is typically still possible by pulling back the equation onto the reference space and using 0 the boundedness of Pn .

Existence, uniqueness, and stability of weak solutions For clarity of the argument, we proceed with several separate results. We start by identifying the limits of the approximating sequences. The bounds in ′ p ∞ p (8.3)–(8.4) give the existence u ∈ LX ∩ LH , z ∈ H(T ) and χ ∈ LX∗ such that, up to a subsequence,

′ ∗ ∞ p p un ⇀u in LH , un ⇀u in LX , un(T ) ⇀z in H(T ), and Aun ⇀χ in LX∗ . We use these to pass to the limit in the approximating equations (8.2). ′ Wp,p ∗ 0 Proposition 8.6. The limit function u ∈ (X,X ) ∩ CH satisﬁes

′ p u˙ + χ +Λu = f in LX∗ , u(0) = u0, u(T )= z.

p,p Proof. For any v ∈ W (Vn, Vn) we can integrate by parts in (8.2) and put the time derivative onto the test function: d (u (t), v(t)) + hAu (t), v(t)i ∗ = hf(t), v(t)i ∗ + (u (t), v˙(t)) . dt n H(t) n X (t), X(t) X (t), X(t) n H(t) t 1 Wp,p For j ≤ n, take v(t) = ψ(t)wj with ψ ∈ C ([0,T ]), which clearly satisﬁes v ∈ (Vn, Vn). Integrating over time and then passing to the limit n → ∞, we obtain

T T T t t (z, ψ(T )w ) − (u , ψ(0)w ) + χ(t), ψ(t)w ∗ = f(t), ψ(t)w ∗ j H(T ) 0 j H0 j X (t), X(t) j X (t), X(t) Z0 Z0 (8.5) T ′ t + (u(t), ψ (t)wj )H(t). Z0 0 R n 0 Since {wj } is a basis for X0, given v ∈ X0, there exist coeﬃcients aj ∈ and a sequence vn = j=1 aj wj such that v → v in X . Hence φ v = n a wt converges to φ v in X(t). Multiplying the above displayed n 0 t n j=1 j j t P equality by aj and summing up j =1, ..., n gives P T T

∗ ∗ (z, ψ(T )φT vn)H(T ) − (u0, ψ(0)vn)H0 + hχ(t), ψ(t)φtvniX (t), X(t) = hf(t), ψ(t)φtvniX (t), X(t) Z0 Z0 T ′ + (u(t), ψ (t)φtvn)H(t). Z0 Take furthermore ψ ∈ D(0,T ). Passing to the limit n → ∞ by using the dominated convergence theorem, we obtain T T T ′ hχ(t), ψ(t)φtviX∗(t), X(t) = hf(t), ψ(t)φtviX∗(t), X(t) + (u(t), ψ (t)φtv)H(t). Z0 Z0 Z0 This is exactly the statement d (u(t), φ v) = hf(t) − χ(t), φ vi ∗ ∀v ∈ X . dt t H(t) t X (t),X(t) 0

44 ′ Hence, by the characterisation oﬀered in Proposition 5.4, it follows that u ∈ Wp,p (X,X∗) with

u˙ +Λu − χ = f

′ . as desired. The fact that u ∈ C0 follows from the continuous embedding Wp,p (X,X∗) ֒→ C0 H ′ H To check the initial condition, let v ∈ Wp,p (X,X∗). Using the transport formula in Theorem 4.21 and the equation for u, we have

∗ ∗ (u(T ), v(T ))H(T ) − (u(0), v(0))H0 = hu˙(t)+Λ(t)u(t), v(t)iX (t), X(t) + hv˙(t),u(t)iX (t), X(t) Z0 T

= hv˙(t),u(t)iX∗(t), X(t) + hf(t) − χ(t), v(t)iX∗(t), X(t) . Z0 t N 1 Taking v(t)= ψ(t)wj for arbitrary j ∈ and ψ ∈ C ([0,T ]), this becomes

T T ′ t t (u(T ), ψ(T )w ) − (u(0), ψ(0)w ) = ψ (t)w ,u(t) ∗ + f(t) − χ(t), ψ(t)w ∗ . j H(T ) j H0 j X (t),X(t) j X (t),X(t) Z0

Comparing this with (8.5), we see that

T T (u(T ), ψ(T )wj )H(T ) − (u(0), ψ(0)wj )H0 = (z, ψ(T )wj )H(T ) − (u0, ψ(0)wj )H0 .

0 Picking ψ such that ψ(T ) = 0 removes the first term on both sides and then the density of {wj } in H0 implies that u(0) = u0. A similar argument gives the final condition. The final step for existence is now to identify the nonlinear term χ in the equation. In the classical setting, the proof of the statement relies on the monotonicity of the operator A, see for example [60, Lemma 30.6]. In our case, the presence of Λ in the equation (and integration by parts formulae) means that in general, the elliptic operator A + Λ is non-monotone. However, as Λ is a lower order term, we are able to mitigate its effects by using an exponential scaling trick.

′ p Proposition 8.7. We have χ = Au in LX∗ . Proof. Let us deﬁne, for γ > 0 to be chosen later, the functions

−γt −γt v(t)= e u(t) and vn(t)= e un(t).

−γt ∞ p Since e belongs to L (0,T ), vn ∈ LVn and we have

∗ ∞ p vn ⇀ v in LH and vn ⇀ v in LX .

−γt −γt γt Deﬁne also χγ (t)= e χ(t) and Aγ (t)ξ = e A(t)e ξ, which is still a monotone operator. Noting that

−γt −γt v˙n(t)= −γvn(t)+ e u˙ n(t) andv ˙(t)= −γv(t)+ e u˙(t), it follows that the new approximations (vn)n and the function v satisfy

p ∗ ∗ −γt ∗ hv˙n + Aγ vn, ηiX (t), X(t) + h(γ + Λ)vn, ηiX (t),X(t) = he f, viX (t), X(t) ∀η ∈ LVn , p ∗ ∗ −γt ∗ hv˙ + χγ , ηiX (t), X(t) + h(γ + Λ)v, ηiX (t),X(t) = he f, viX (t), X(t) ∀η ∈ LX . (8.6) Now deﬁne ∗ 1 L (t): X(t) → X (t) by hL (t)v, ηi ∗ = h(2γ + Λ)v, ηi ∗ . γ γ X (t), X(t) 2 X (t),X(t)

We choose the constant γ in such a way that Lγ is monotone (in the case of the p-Laplace equation, any γ satisfying 2γ ≥ k∇g · wkL∞ works, and in general such a choice is possible due to Assumption 4.5 (iii), see also the third condition in Remark 4.13). Now, on the one hand, testing (8.6) with η = v leads to

1 d 2 −γt kvk + hχ + L v, vi ∗ = e f, v ∗ , 2 dt H(t) γ γ X (t), X(t) X (t), X(t) which we integrate over [0,T ] to obtain

T T ku k2 e−γT ku(T )k2 −γt 0 H0 H(T ) hχ + L v, vi ∗ = e f, v ∗ + − . (8.7) γ γ X (t), X(t) X (t), X(t) 2 2 Z0 Z0

45 On the other hand, the same calculation for the approximation vn now gives

T T 2 −γT 2 kP u k e kun(T )k −γt n 0 H0 H(T ) h(L + A )v , v i ∗ = e f, v ∗ + − , γ γ n n X (t), X(t) n X (t), X(t) 2 2 Z0 Z0 whence taking the limit superior and using the weak lower-semicontinuity of norms, we obtain

T T ku k2 e−γT ku(T )k2 −γt 0 H0 H(T ) lim sup h(A + L )v ), v i ∗ ≤ e f, v ∗ + − . (8.8) γ γ n n X (t), X(t) X (t), X(t) 2 2 n Z0 Z0

Combining (8.7) with (8.8) then gives

T T

hχγ + Lγv, viX∗(t), X(t) ≥ lim sup h(Aγ + Lγ)vn, vniX∗(t), X(t) . (8.9) Z0 n Z0 p Now take an arbitrary η ∈ LX . Monotonicity of Aγ +Lγ implies that h(Aγ + Lγ )(vn) − (Aγ + Lγ )(η), vn − ηi≥ 0, which we can expand to obtain

h(Aγ + Lγ )vn, vniX∗(t), X(t) ≥ h(Aγ + Lγ )vn, ηiX∗(t), X(t) + h(Aγ + Lγ )η, vn − ηiX∗(t), X(t) . Z0 Taking the limit superior and using (8.9) on the left-hand side and the convergence results on the right-hand side, we get

hχγ + Lγ v, viX∗(t), X(t) ≥ hχγ + Lγ v, ηiX∗(t), X(t) + h(Aγ + Lγ )η, v − ηiX∗(t), X(t) . Z0 This reads

hχγ + Lγ v − Aγ η − Lγ η, v − ηiX(t), X∗(t) ≥ 0. Z0 To conclude the proof we apply the well-known Minty’s monotonicity trick, which gives χγ = Aγ v and hence χ = Au. All in all, combining the previous results shows that the limit function u is indeed a weak solution as per Deﬁnition 8.1. Finally, the result below establishes stability of solutions with respect to initial conditions and uniqueness follows as a consequence, concluding the proof of Theorem 8.2.

Proposition 8.8. If u1 and u2 are weak solutions of (8.1) corresponding to initial data u10 and u20, then

Cwt/2 ku1(t) − u2(t)kH(t) ≤ e ku10 − u20kH0 .

In particular, weak solutions are unique.

Proof. By testing the equation for both u1 and u2 with v = u1 − u2 and subtracting we obtain

1 d 2 Cw 2 ku − u k + hAu − Au ,u − u i ∗ ≤ ku − u k . 2 dt 1 2 H(t) 1 2 1 2 X (t), X(t) 2 1 2 H(t) Monotonicity of A implies that we can neglect the second term on the left-hand side and then Gronwall’s inequality gives the result.

A Technical results

0 0 0 −1 0 −⊺ Lemma A.1. The derivative of At = Jt (DΦt ) (DΦt ) satisﬁes

0 0 0 0 −1 ⊺ 0 −⊺ ∂tAt = φ−t(∇ · w(t))At − Jt (DΦt ) (φ−t(Dw(t)) + (φ−t(Dw(t))) )(DΦt ) .

0 Proof. To ease presentation, we deﬁne Dt = DΦt . We begin with

0 0 −1 −⊺ 0 −1 −⊺ ∂tAt = Jt φ−t(∇ · w(t))(Dt) (Dt) + Jt ∂t((Dt) (Dt) ). To simplify the second term, using the formula (M −1)′ = −M −1M ′M −1 for diﬀerentiating the inverse of a matrix M, and the identity ∂tDt = φ−t(Dw(t))Dt,

46 we get

−1 −⊺ −1 −1 −⊺ −1 −1 −1 ⊺ ∂t((Dt) (Dt) )= −(Dt) ∂t(Dt)(Dt) (Dt) − (Dt) ((Dt) ∂t(Dt)(Dt) ) −1 −1 −⊺ −1 −1 −1 ⊺ = −(Dt) φ−t(Dw(t))Dt(Dt) (Dt) − (Dt) ((Dt) φ−t(Dw(t))Dt(Dt) ) −1 −⊺ −1 ⊺ −⊺ = −(Dt) φ−t(Dw(t))(Dt) − (Dt) (φ−t(Dw(t))) (Dt) .

Hence

0 0 −1 −⊺ 0 −1 −⊺ −1 ⊺ −⊺ ∂tAt = Jt φ−t(∇ · w(t))(Dt) (Dt) − Jt ((Dt) φ−t(Dw(t))(Dt) + (Dt) (φ−t(Dw(t))) (Dt) ) 0 0 −1 ⊺ −⊺ = φ−t(∇ · w(t))At − Jt (Dt) (φ−t(Dw(t)) + (φ−t(Dw(t))) )(Dt) .

0 Let us now give an expression ∂tAt and see how it acts. 1 Lemma A.2. For v ∈ H0 (Ω(t)), we have the identity

t ⊺ t −1 0 t −⊺ ⊺ ⊺ ⊺ J0∇v(t) (DΦ0) φt(∂tAt )(DΦ0) ∇ψ = ∇v(t) ∇ψ∇ · w(t) − ∇v(t) (Dw(t) + (Dw(t)) )∇ψ ZΩ(t) ZΩ(t) 1 for all ψ ∈ H0 (Γ(t)). 0 T Proof. The formula φt(∇v˜)= φt(DΦt ) ∇v and Lemma A.1 allows us to write

⊺ 0 ⊺ 0 ∇v˜(t) ∂tAt ∇ϕ = ∇v˜(t) φ−t(∇ · w(t))At ∇ϕ ZΩ0 ZΩ0 ⊺ 0 0 −1 ⊺ 0 −⊺ − ∇v˜(t) Jt (DΦt ) (φ−t(Dw(t)) + (φ−t(Dw(t))) )(DΦt ) ∇ϕ ZΩ0 = ∇v(t)⊺∇ψ∇ · w(t) ZΩ(t) ⊺ 0 0 −1 ⊺ 0 −⊺ 0 ⊺ − ∇v(t) φt(DΦt )φt(DΦt ) (Dw(t)+ Dw(t)) )φt(DΦt ) φt(DΦt ) ∇ψ ZΩ(t) = ∇v(t)⊺∇ψ∇ · w(t) − ∇v(t)⊺(Dw(t) + (Dw(t))⊺)∇ψ. ZΩ(t) n n Proof of Lemma 8.4. Denoting the solution vector Un(t) = (u1 (t),...,un(t)), the linear terms

t t t t t B(t) = (w , w ) , G(t) = λ(t; w , w ), F (t) = f(t), w ∗ , ij i j H(t) ij i j j j X (t), X(t) and the nonlinear term

t A(t; U (t)) = A(t)u (t), w ∗ , n j n j X (t), X(t) the problem (8.2) is equivalent to the system of ODEs

B(t)U˙ n(t)+ A(t; Un(t)) + G(t)Un(t)= F (t),

Un(0) = (α1,...,αn),

n 0 where {αj} are the coefficients of Pnu0 = j=1 αj wj . Since B(t) is a Gram matrix (and hence invertible) and the operators defining the lower-order terms are measurable in time and continuous in ‘space’, the conclusion follows from the classical Carathéodory existenceP theory.

Acknowledgements

AA was partially supported by the DFG through the DFG SPP 1962 Priority Programme Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization within project 10. ADj was supported by the DFG under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ and CRC 1114 “Scaling Cascades in Complex Systems”.

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