Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. However, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure- valued weak versus strong uniqueness using the averaged relative entropy inequality.


Introduction
For systems of hyperbolic conservation laws, the class of measure-valued solutions [16] provides a notion of solvability vast enough to support a global existence theory. These solutions usually arise as limits of converging sequences satisfying an approximating parabolic problem [12]. As these solutions are considered to be very weak, it is crucial to examine their stability properties with respect to classical solutions and to attempt that in their natural energy framework. The relative entropy method of Dafermos [11,10] and DiPerna [15] provides an analytical framework upon which one can examine such questions, and has been tested in a variety of contexts (e.g. [5,14,20,8,18,6]).
In this article, we derive (in the appendix) a framework of dissipative measure valued solutions for the system of adiabatic polyconvex thermoelasticity, motivated by approximating that system by the system of thermoviscoelasticity on the natural energy framework. The relative entropy method is then used to show weak-strong uniqueness for polyconvex thermoelasticity in the class of measure-valued solutions. The main novelty of this work is the derivation of the averaged relative entropy inequality with respect to a dissipative measure-valued solution. This solution is defined by means of generalized Young measures, describing both oscillatory and concentration effects. The analysis is based on the embedding of polyconvex thermoelasticity into an augmented, symmetrizable, hyperbolic system, [7]. However, the embedding cannot be used in a direct manner, and notably, instead of working with the extended variables, we base our analysis on the parent system in the original variables using the weak stability properties of some transport-stretching identities, which allow us to carry out the calculations by placing minimal regularity assumptions in the energy framework.
Consider the system of adiabatic thermoelasticity, describing the evolution of a thermomechanical process y(x, t), θ(x, t) ∈ R 3 × R + with (x, t) ∈ R 3 × R + . Here, F ∈ M 3×3 stands for the deformation gradient, F = ∇y, while v = ∂ t y is the velocity of the motion y and θ is the temperature. The condition imposes that F is a gradient and comes from equation (1.1) 1 as an involution inherited from the initial data. The stress is denoted as Σ iα , the internal energy as e and the radiative heat supply as r. The requirement of consistency with the Clausius-Duhem inequality imposes that the elastic stresses Σ, the entropy η and the internal energy e are related to the free-energy function ψ via the constitutive theory ψ = ψ(F, θ), Σ = ∂ψ ∂F , η = − ∂ψ ∂θ , e = ψ + θη .
The main result of this article is the weak-strong uniqueness of polyconvex adiabatic thermoelasticity (1.1)- (1.4) in the class of dissipative measure-valued solutions. The advantage of the dissipative framework is that the averaged energy equation holds in its integrated form. Even though this notion of solutions is generally considered to be very weak, not possessing detailed information, this result contributes to a long list of similar works [1,17,13,14,8] on hyperbolic systems of conservation laws, pointing out the importance of this framework in the analysis of such physical problems. Unlike the case of scalar conservation laws [16,22], where the theory of Young measures suffices to deal with nonlinearities and overcome oscillatory behaviors, when it comes to hyperbolic systems, one must take into account the formation of both oscillations and concentrations. In our case, the concentration effects are described through a concentration measure, which appears in the energy equation since the Fundamental Lemma of Young measures cannot represent the weak limits of 1 2 |v| 2 + e, due to lack of L 1 precompactness. This is illustrated in Appendix A. Thus we turn our attention to the theory of generalized Young measures [1,17,5,14,8,20] and apply the relative entropy formulation to compare a dissipative measure-valued solution to polyconvex thermoelasticity against a strong solution.
We organize this paper as follows: In Section 2, we define the notion of dissipative measure-valued solutions for polyconvex thermoelasticity. This definition comes as a result of the limiting process we discuss in Appendix A, starting from the associated viscous problem. Section 3 is dedicated to the study of the generated Young measure and the concentration measure, which is a well-defined, nonnegative Radon measure for a subsequence of approximate solutions coming from a uniform bound on the energy. In Section 4 we calculate the averaged relative entropy inequality (4.16) and in Section 5 we use it to prove the main theorem on uniqueness of strong solutions in the class of measure-valued solutions. The proof is heavily based on the estimates (5.2) and (5.4)-(5.7) on the relative entropy, namely Lemmas 5.1, 5.2, which are stated and proved at the level of the original variables, instead of the extended ones. As a result, we only assume quite minimal growth hypotheses on the constitutive functions, which guarantee all the necessary technical requirements for the dissipative measure-valued versus strong uniqueness to hold. Additionally, the proof is carried on with respect to a dissipative solution which satisfies an averaged and integrated version of the energy equation, where the concentration measure appears. This setting has the strong advantage that we need no artificial integrability restrictions on the energy equation. Similar results are available for the incompressible Euler equations [5], for polyconvex elastodynamics [14], and for the isothermal gas dynamics system [20].

Measure-valued solutions for polyconvex adiabatic thermoelasticity
Consider the system of adiabatic thermoelasticity (1.1), (1.2) together with the entropy production identity for any deformation gradient F and velocity field v. Additionally, the stress tensor Σ becomes The first nine components of Φ(F ) are the components of F, therefore (1.3) implies that we can express the entropy η and the internal energy e with respect to the null-Lagrangian vector Φ(F ), namely where we have setη This allows to supplement the equations of polyconvex thermoelasticity (1.1), (1.2) with (2.3) and write while the entropy production identity (2.1) becomes This implies that (ξ = Φ(F ), v, θ) satisfies the augmented system that consists of conservation laws in R 23 subject to the involution (2.9) 4 . The augmented system satisfies the entropy production identity and is thus symmetrizable; see [7] for further details.
The system (2.9) belongs to a general class of hyperbolic conservation laws of the form [8]. Due to (2.10) it is symmetrizable, hyperbolic in the extended variables. A general theory including a theorem establishing recovery of classical solutions from dissipative measure-valued solutions for hyperbolic systems endowed with a convex entropy, was developed in [8]. We note that since in the variables (F, v, θ) system (2.7) is not equipped with a convex entropy, we cannot treat this problem as a direct application of the general setting developed in [8]. In [7], system (1.1)-(1.5) was studied by augmenting it to (2.7) using the relative entropy method in order to prove convergence from thermoviscoelasticity to the system (1.1)-(1.5). The objective in the present paper is to prove a weak-strong uniqueness theorem in the context of measure-valued solutions. This requires to work at the level of the original rather than the augmented system what presents various technical challenges. Following the theory on generalized Young measures [1,8,17], we define a dissipative measure-valued solution to polyconvex thermoelasticity, which involves a parametrized Young measure ν = ν (x,t) describing the oscillatory behavior of the solution and a Radon measure γ ∈ M + (Q T ) describing concentration effects. According to the analysis in Appendix A, we can treat dissipative measure-valued solutions as limits of an approximating solution for the associated viscous problem, that satisfy an averaged and integrated energy equation. The reason behind the formation of concentrations, lies with the fact that the energy function (x, t) → |v| 2 + e(F, θ) is not weakly precompact in L 1 and thus, the Young measure representation fails. Since the only uniform bound at one's disposal is on the energy, the way we construct these solutions corresponds to a minimal framework obtained from this natural bound, for viscosity approximations of the adiabatic thermoelasticity system. The analysis in Appendix A leads to the following definition: Definition 2.1. A dissipative measure valued solution to polyconvex thermoelasticity (2.7), (2.8) consists of a thermomechanical process (y(t, x), θ(t, x)) : a parametrized family of probability Young measures ν = ν (x,t)∈Q T , with averages and a nonnegative Radon measure γ ∈ M + (Q T ), where p ≥ 4, q ≥ 2, ρ > 1, ℓ > 1, which satisfy the averaged equations in the sense of distributions, together with the integrated form of the averaged energy equation, In this definition, the first equation (2.13) 1 holds in a classical weak sense under the regularity conditions (2.12),(2.11) placed on the motion and its derivatives for p ≥ 4, q ≥ 2, ρ, ℓ > 1, as a consequence of the weak continuity of the null-Lagrangian vector (F, cofF, det F ) and the weak continuity of the transport-stretching identities We summarize the corresponding results, taken out of [3] and [13], in the following lemma. As the weak continuity property is important for the forthcoming analysis, we present the proof here for the reader's convenience.
Proof. We note the formulas, for smooth maps, and Step 1. For y ∈ W 1,∞ (L 2 (T 3 ))∩L ∞ (W 1,p (T 3 )), we extend y to a function defined for all times, by putting y(t, x) = y(0, x), for t ≤ 0. The extended y belongs to the same regularity class. Define the convolution (in space and time) and such that for all s < ∞, T > 0 : Let F ǫ = ∇y ǫ and v ǫ = ∂ t y ǫ . Since the cofactor matrix is bilinear in the components of F, and the determinant is trilinear, it follows by repeated use of Hölder inequalities that for some numerical constant C, We thus conclude: Passing to the limit ǫ → 0, for p ≥ 4, in the formulas we obtain (2.15) in the sense of distributions and complete the proof of (i).
Step 2. Let {y ε } ε>0 be a family satisfying the uniform bound (2.16) and let F ε = ∇y ε and v ε = ∂ t y ε . We adapt the proof of [3, Lemma 6.1] suggesting to write the cofactor and the determinant in divergence form: With p ≥ 2, q ≥ p p−1 , hypothesis (2.16) implies y ε ⇀ y weakly in W 1,2 loc ([0, ∞) × T 3 ) along subsequences and Rellich's theorem (for dimension 3 + 1) implies y ε → y strongly 3 the dual exponent to 4. Therefore, we can pass to the limit in the sense of distributions: and similarly for the determinant since cofF ε ⇀ cofF weak- * in L ∞ (L q ), for q > 4 3 the dual exponent to 4. The distributional limits in (2.18) and (2.19) coincide with the limits in the weak- * topology. Altogether we have Using the weak continuity properties of cofF and det F and that for p ≥ 4 equations (2.15) hold for functions y of class (2.16), we conclude that equations (2.15) are weakly stable.
Remark 2.1. On the definition of the dissipative measure-valued solution: 1. Combining the requirements of Lemma 2.1, with those of Lemmas 5.1 and 5.2, we must assume the exponents p ≥ 4, q ≥ 2, ρ, ℓ > 1. 2. Henceforth, we assume the measure γ 0 = 0, meaning that we consider initial data with no concentrations at time t = 0. 3. Next, we highlight why we choose to work with the system in the physical variables (Φ(F ), v, θ) instead of the extended ones (ξ, v, θ): This allows to avoid imposing restrictive growth conditions on the constitutive functions with respect to the cofactor and the determinant derivatives. From previous works in isothermal polyconvex elastodynamics (e.g. [14]) or even in [7], it becomes evident that when considering the extended system, one has to impose growth condition on terms where ξ = (F, ζ, w), in order to achieve representation of the associated weak limits via Young measures. The resulting regularity class of functions is far too restrictive and in particular functions with general power-like behavior do not satisfy such assumptions and their weak-limits cannot be represented. By contrast, if one works with the original variables, the growth hypotheses (3.1)-(3.2) placed on ψ(F, θ) and e(F, θ), which are compatible with the constitutive theory, are also sufficient to allow representation of the corresponding weak limits. 4. The reasoning behind studying the integrated form of the averaged energy equation lies in the technical advantage that, one does not need to place any integrability condition on the right hand-side of the energy equation (2.7) 3 , namely on the term since it appears as a divergence and its contribution integrates to zero.

Young measures and concentration measures
We assume the following growth conditions on the constitutive functions e(F, θ), ψ(F, θ), η(F, θ) and Σ(F, θ) : -which are consistent with the constitutive theory (1.3)-for some constant c > 0 and p ≥ 4, ℓ > 1. As presented in the appendix, we consider measure-valued solutions as limits of approximations that satisfy the uniform bound coming from the energy conservation equation (2.7) 3 , given that the radiative heat supply r is a bounded function in L 1 (Q T ). The growth condition (3.1) in combination with (3.5) suggests that the functions F ε ∈ L p , v ε ∈ L 2 , θ ε ∈ L ℓ are all (uniformly) bounded in the respective spaces. The approximating sequence U ε = (F ε , v ε , θ ε ), represents weak limits of the form and its action is well-defined for all functions f that grow slower than the energy. To take into account the formation of concentration effects, we introduce the concentration measure γ, depending on the total energy. This is a well-defined nonnegative Radon measure for a subsequence of e(F ε , θ ε ) + 1 2 |v ε | 2 . To prove this claim, let us define the sets Let X be a locally compact Hausdorff space, where we define the set of all Radon measures M(X) and all positive Radon measures M + (X), while Prob(X) denotes all probability measures on X. Let Ω be any open subset of R d and fix a Radon measure λ on Ω. We denote by P(λ; X) = L ∞ w (dλ; Prob(X)) the parametrized families of probability measures (ν z ) z∈Ω acting on X which are weakly measurable with respect to z ∈ Ω. When λ is the Lebesgue measure, we use the notation P(λ; X) = P(Ω; X).
The following theorem as it appears in [1,17] uses the theory of generalized Young measures to describe weak limits of the form for φ ∈ C 0 (Ω), any bounded sequence u n in L 1 , and test functions g such that (3.7) There exists a subsequence {u n k }, a nonnegative Radon measure µ ∈ M + (Ω) and parametrized families of probability measures for any g ∈ F 1 .
Given that the only available bound for the approximate sequence U ε is of the form we want to represent the weak limits wk- * lim In order to apply Theorem 3.1, we perform the change of variables imposing that the function g grows like Then Theorem 3.1 applies to represent the wk- * limits of g: Then, property (3.7) implies that and Therefore, given the bound (3.5) and assuming that the recession function exists and is continuous for all (|F | p−1 F, |v|v, θ ℓ ) ∈ S d 2 +d ∩ {c > 0}, we have that (along a subsequence) (3.8)

The averaged relative entropy inequality
The augmented system (2.9) belongs to a general class of hyperbolic systems of the form where U = U (x, t) ∈ R n , is the unknown with x ∈ R d , t ∈ R + and A, f α : R n → R n are given smooth functions of U. It is symmetrizable in the sense of Friedrichs and Lax [19], under appropriate hypotheses: The map A(U ) is globally invertible and there exists an entropy-entropy flux pair (H, q), i.e. there exists a smooth multiplier G(U ) : R n → R n such that In our case while the (mathematical) entropy is given by H(U ) = −η(Φ(F ), θ), the entropy flux q α = 0 and the associated multiplier is see [8,7]. Consider a strong solution (Φ(F ),v,θ) T ∈ W 1,∞ (Q T ) to (2.7) that satisfies the entropy identity (2.8) and a dissipative measure valued solution to (2.7), (2.8) according to Definition 2.1. We write the difference of the weak form of equations (2.7), (2.8) and (2.13), (2.14) to obtain the following three integral identities and ν, for any φ i ∈ C 1 c (Q T ), i = 1, 2 and φ 3 ∈ C 1 c [0, T ). Similarly, testing the difference of (2.8) and (2.13) 3 against φ 4 ∈ C 1 c (Q T ), with φ 4 ≥ 0, we have and ν, For inequality (4.4), we choose accordingly (4.8) Adding together (4.5), (4.6), (4.7) and (4.8), we obtain the integral inequality Using the entropy identity (2.8) and the null-Lagrangian property (2.2), the quantity K(x, t) in the integrand on the right hand-side of (4.9) becomes employing (2.7) 1 and (2.2). Here, we use the quantites and ν, Next, we rewrite the terms and because of the null-Lagrangian property (2.2). Also, we observe that Finally, if we define the averaged quantity and then combine (4.9),(4.10),(4.13) and (4.14), we arrive at the relative entropy inequality (4. 16) We note that the last term in (4.13) vanishes when we substitute into the integral relation (4.9).

Uniqueness of smooth solutions in the class of dissipative measure-valued solutions
In this section, we state and prove the main theorem on dissipative measure-valued versus strong uniqueness. Before we proceed with the proof, we show some useful estimates on the terms appearing in (4.16), which yield the relative entropy as a "metric" measuring the distance between the two solutions. These bounds are obtained by using the convexity of the free energy function in the compact domain and the growth conditions
In the complementary region |F | p +θ ℓ +|v| 2 ≤ R, observe that (F, cofF, det F, v, θ) takes values in the set for some constant C. We use the convexity of the entropyH(V ) in the symmetric variables sinceH(V ) is convex in V and D * is the compact domain determined by the map V = A(U ) and the set D defined above. Moreover, using the invertibility at the map U → A(U ) Therefore for K 2 := δ min{H V V (V * )} > 0 and the proof is complete. for some positive constant c, the following bounds hold true: (i) There exist constants C 1 , C 2 , C 3 , C 4 > 0 such that and for all (F ,v,θ) ∈ Γ M,δ . (ii) There exist constants K ′ 1 , K ′ 2 and R > 0 sufficiently large such that Proof. We divide the proof into 5 steps.
Step 3. We proceed in a similar manner as in Step 2. to prove (5.6). First we study the region |F | p + |v| 2 + θ ℓ > R and we use growth assumption (3.3) and relation (2.5) to get So immediately we deduce for R large enough. On the complementary region |F | p + |v| 2 + θ ℓ ≤ R, by (5.2). Choosing C 3 = max{C ′ 3 , C ′′ 3 }, the proof of (5.6) is complete.
We now consider a dissipative measure-valued solution for polyconvex thermoelasticity as defined in Definition 2.1. Using the averaged relative entropy inequality (4.16), we prove that in the presence of a classical solution, given that the associated Young measure is initially a Dirac mass, the dissipative measure-valued solution must coincide with the classical one. Proof. Let {ϕ n } be a sequence of monotone decreasing functions such that ϕ n ≥ 0, for all n ∈ N, converging as n → ∞ to the Lipschitz function for some ε > 0. Writing the relative entropy inequality (4.16) for r(x, t) =r(x, t) = 0, tested against the functions ϕ n we have (5.9) Passing to the limit as n → ∞ we get Passing now to the limit as ε → 0 + and using the fact that γ ≥ 0 in combination with the estimates (5.4), (5.5), (5.6) and (5.7), we arrive at for t ∈ (0, T ). Note that the constant C depends only on the smooth bounded solutionŪ . Then Gronwall's inequality implies and the proof is complete by (5.8).
An extension of Theorem 5.1 holds in case we assume r(x, t) =r(x, t) = 0. For this purpose, we need the additional assumption (5.10) to control the terms that arise from the radiative heat supply in (4.16). We first prove the following lemma: for some small positive constant δ. Then there exists a constant C 5 > 0 such that ν, Proof. Assume first that |F | p + θ ℓ + |v| 2 > R. Then ν, Choosing R sufficiently large, we get for ℓ > 1 ν, where, the last inequality holds because of Lemma 5.1 and the constant C ′ 5 depends on r, δ, M and δ.
again by estimate (5.2). By choosing the proof is complete.
Proof. The proof is a simple variant of the one for Theorem 5.1. Assuming the sequence {ϕ n } as before, the relative entropy inequality (4.16) becomes Passing to the limit as n → ∞ and then as ε → 0 + we obtain for t ∈ (0, T ). Here, we used that γ ≥ 0 and the estimates (5.4), (5.5), (5.6), (5.7) and (5.11), so that constant C depends on the smooth bounded solutionŪ and δ. By virtue of Gronwall's inequality and (5.8), we conclude the proof.
Let us note that, as Lemma 5.3 indicates, one needs to assume (5.10) in order to be able to bound from below the averaged temperature ν, λ θ and achieve estimate (5.11). Though it could be considered as a rather mild assumption, it is interesting that all the estimates in Lemmas 5.1 and 5.2 that involve the averaged temperature, do not require (5.10) to hold. This is because the averaged temperature is involved only through the constitutive functionsψ,ê andη which we assume to be smooth enough, i.e.ψ =ê − θη ∈ C 2 , and therefore we avoid any loss of smoothness as the temperature approaches zero.
Appendix A. The natural bounds of viscous approximation for polyconvex thermoelasticity Since measure-valued solutions usually occur as limits of an approximating problem, consider the system of polyconvex thermoelasticity with Newtonian viscosity and Fourier heat conduction Suppose first that the energy radiation r ≡ 0. The viscosity and heat conduction coefficients are assumed to satisfy the condition for some constants µ 0 , k 0 > 0, we are going to examine how we can obtain measure-valued solutions in the limit as µ 0 → 0 and k 0 → 0. We work in a periodic domain in space We impose the growth conditions for some constant c > 0 and p, ℓ > 1. Integrating the energy equation (A.1) 3 in Q T we get

Therefore (A.3) implies that
and then the functions (F µ,k , v µ,k , θ µ,k ) are all bounded in the spaces: and weakly converging to the averages Integrating now (A.1) 4 (r ≡ 0), in Q T we obtain (A.10) Then (A.5) and (A.9) immediately imply thatη(Φ(F µ,k ), θ µ,k ) ∈ L ∞ (L 1 ) while Now, let us consider the first equation in (A.1). We employ Lemma 2.1, in order to pass to the limit in the minors and the identities (2.15). It follows that (A.1) 1 holds in the classical weak sense for motions with regularity as in (2.16) and with p ≥ 4, q ≥ 2, ρ, ℓ > 1.
To pass to the limit in the second equation (A.1) 2 , we use the Theorem of Ball [2] on representation via Young measures in the L p setting: Lemma A.1. Let f ǫ :Q T → R m be a bounded function in L p . Then for all F : R m → R which are continuous and such that F (f ǫ ) is L 1 weakly precompact, there holds (along a subsequence) F (f ǫ ) ⇀ ν, F , weakly in L 1 (Q T ).
If f ǫ :Q T → R m is uniformly bounded in Q T , then for all continuous F : R m → R there holds (along a subsequence) F (f ǫ ) ⇀ ν, F , weak- * in L ∞ (Q T ).
Remark A.1. Testing the energy equation (A.1) 3 against a test function ϕ = ϕ(x, t) ∈ C 1 c (Q T ), yields a different notion of measure-valued solution in the limit, the so-called entropy measure-valued solution. In that case, additional assumptions are required. First, one should represent the term ∂ψ ∂ξ B (Φ(F µ,k ), θ µ,k )