Thermoviscoelasticity in Kelvin–Voigt Rheology at Large Strains

The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration by using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back to the reference configuration. The existence of weak solutions in the quasistatic setting, that is inertial forces are ignored, is shown by time discretization.


Introduction
For a long time, thermoviscoelasticity was considered as a quite difficult problem even at small strains, mainly because of the nonlinear coupling with the heattransfer equation which has no obvious variational structure; hence special techniques had to be developed. It took about two decades after the pioneering work by Dafermos [14] in one space dimension that first three-dimensional studies occurred (cf. for example [7,12,43]). The basic new ingredient was the L 1 -theory for the nonlinear heat equation developed in [9,11]. At large strains, in simple materials, the problem is still recognized to be very difficult even for the case of mere viscoelasticity without coupling with temperature, and only few results are available if the physically relevant frame-indifference is respected, as articulated by Ball [2], see also [3,4]. In particular, local-in-time existence [27] or existence of measure-valued solutions [15,18]  in this direction are [52] for a general three-dimensional theory, but not respecting frame indifference and the determinant constraints, or [34] for a one-dimensional theory using the variational structure. While the static theory for large-strain elasticity developed rapidly after [2], there are still only few result for time-dependent processes respecting frame indifference as well as the determinant constraint. The first cases were restricted to rate-independent processes, such as elastoplasticity (cf. [31,36]) or crack growth (cf. [16], see [35,Sec. 4.2] for a survey. Recently the case of viscoplasticity was treated in [38]. The main features of the model discussed in this work can be summarized in brief as follows: the thermoviscoelastic continuum is formulated at large strains in a reference configuration, that is the Lagrangian approach. The concepts of 2nd-grade nonsimple material is used, which gives higher regularity of the deformation. The heat transfer is modeled by the Fourier law in the actual deformed configuration, but transformed (pulled back) into the reference configuration for the analysis. Our model respects both static frame-indifference of the free energy and dynamic frame indifference for the dissipation potential. Moreover, the local non-selfpenetration is realized by imposing a blowup of the free energy if the determinant of the deformation gradient approaches 0 from above, however we do not enforce global non-selfpenetration. Also, we neglect inertial effects; cf. Remark 6.6 for more detailed discussion.
Let us highlight the important aspects of the presented model and their consequences: (α) The temperature-dependence of the free energy creates adiabatic effects involving the rate of the deformation gradient. To handle this, the Kelvin-Voigt-type viscosity is used to control the rate of the deformation gradient. In addition, we separate the purely mechanical part, cf. (2.15) below, which allows us to decouple the singularities of large-strain elasticity from the heat equation. (β) The heat transfer itself [and also the viscosity from (α)] is clearly rate dependent and the technique of rate-independent processes supported by a variationally efficient energetic-solution concept cannot be used (which also prevents us from excluding possible global selfpenetration). (γ) The equations for the solid continuum need to be formulated and analyzed in the fixed reference configuration but transport processes (here only the heat transfer) happen rather in the actual configuration and the pull-back procedure needs the determinant of the deformation gradient to be well away from 0. To achieve this, we exploit the concept of 2nd-grade nonsimple materials together with the results of Healey and Krömer [24], which allow us to show that the determinant for the deformation gradient is bounded away from 0, see Section 3.1. (δ) The transport coefficients depend on the deformation gradient because of the reasons in point (γ). For this, measurability in time is needed and thus the concept of global quasistatic minimization of deformation (as in rate-independent systems [35] or in viscoplasticity in [38]) would not be satisfactory; therefore we rather control the time derivative of the deformation, which can be done either by inertia (which is neglected in our work) or by the Kelvin-Voigt-type viscosity from (α).
( ) The viscosity from (α) must satisfy time-dependent frame indifference as explained in [1], thus it is dependent on the rate of the right Cauchy-Green tensor rather than on the rate of the deformation gradient itself. However, the adiabatic heat sources/sinks involve terms where the rate of the deformation gradient occurs directly. To control the latter by the former, we exploit results of Neff [39] in the extension by Pompe [41] for generalized Korn's inequalities, see Section 3.2. Here, again the mentioned concept of 2nd-grade nonsimple materials is used to control the determinant of the deformation gradient, see (γ).
As mentioned above, our model heavily relies on the strain-gradient theories to describe materials, referred as nonsimple, or also multipolar or complex. This concept has been introduced long time ago, cf. [51] or also for example [8,19,28,40,46,50] and in the thermodynamical concept also [6]. In the simplest scenario, which is also used here, the stored-energy density depends only on the strain F = ∇y and on the first gradient ∇ F of the strain. This case is called 2nd-grade nonsimple material. Possible generalization using only certain parts of the 2nd gradient in the spirit of [25] still need to be explored.
The structure of the paper is as follows: in Section 2 we present the model in physical and mathematical terms. After the precise definition of our notion of solution, Theorem 2.2 provides the main existence result for global-in-time solutions for the large-strain thermoviscoelastic system, while Corollary 2.3 gives the corresponding existence result for viscoelasticity at large strain and at constant temperature, which, to the knowledge of the authors, is also new. A related result for isothermal large-strain viscoelasticity is derived in [20], but there the limit of small strains is treated.
After proving some auxiliary results about local invertibility of deformations and the Euler-Lagrange equations, a generalized Korn's inequality, and about Chain rules in Section 3, we start the proof of the main result in Section 4 by introducing certain regularizations as well as a time-incremental approach. This is particularly constructed in such a split (sometimes called staggered) way that the deformation is first updated at fixed temperature from the previous time level and then the temperature is updated, where in some terms the old and in others the new deformation is used. Another important step in the analysis is the usage of an energy-like variable w = w(∇ y, θ) instead of temperature θ , which enables us to exploit the balancelaw structure of the heat equation; cf. [30,32] for arguments for the preference of energy in favor of temperature. After proving existence and a-priori estimates for such approximate solutions in Section 4, we continue by convergence in Section 5 by limiting the time discretization. Thus, as an intermediate result, Proposition 5.1 provides the existence of solutions (y ε , θ ε ) of the regularized problem. Eventually, in Section 6 we finally show that the limit ε k → 0 for (y ε k , θ ε k ) → (y, θ) can be controlled in such a way that (y, θ) are the desired solutions. We conclude with a few remarks concerning potential generalizations and further applications of the methods.

Modeling of Thermoviscoelastic Materials in the Reference Configuration
We will use the Lagrangian approach and formulate the model in the reference (fixed) domain Ω ⊂ R d being bounded with a Lipschitz boundary Γ . In fact, occasionally we will assume Γ smooth in order to reveal the classical formulation of the problem, cf. (2.13)-(2.14) based on the arguments (2.28)-(2.29) below. We assume d 2 although, of course, the rather trivial case d = 1 works too if p 2 is assumed, additionally to p > d, in (2.30) below. We will consider a fixed time horizon T > 0 and use the notation I := [0, T ], Q := I × Ω, and Σ := I × Γ . For readers' convenience, Table 1 summarizes the main nomenclature used throughout the paper.
H strain-gradient energy, Φ cpl coupling energy, Ψ = M + Φ cpl free energy, E = M + W total energy, W thermal energy, ζ potential of dissipative forces, ξ rate of dissipation (=heat production), cf. also [34,52] for the isothermal case and [29] for the general case. The sum of the conservative and the dissipative parts corresponds to the Kelvin-Voigt rheological model in the quasistatic variant (neglecting inertia). The notation " ∂ " is used for partial derivatives (here functional or later in Euclidean spaces), while (·) will occasionally be used for functions of only one variable. Writing (2.2) locally in the classical formulation, one arrives at the nonlinear parabolic 4th-order partial differential equation expressing quasistatic momentum equilibrium where the viscous stress is σ vi = σ vi (F, . F, θ) and the elastic stress is σ el = σ el (F, θ), while h el is a so-called hyperstress arising from the 2nd-grade nonsimple material concept, cf. for example [40,46,51]. In view of the local potentials used in (2.2), we have where G ∈ R d×d×d is a placeholder for ∇ F. An important physical requirement is static and dynamic frame indifference. For the elastic stresses, static frame indifference means that for all R ∈ SO(d), F and G. For the viscous stresses, dynamic frame indifference means that for all smoothly time-varying R: t → R(t) ∈ SO(d), cf. [1]. Note that R may depend on t but not on x ∈ Ω, since frame-indifference relates to superimposing time-dependent rigid-body motions.
In terms of the thermodynamic potentials ζ , ψ, and H , these frame indifferences read as for R, F and ∇ F as above. These frame indifferences imply the existence of reduced potentials ψ, ζ , and H such that where B = G ·G ∈ R (d×d)×(d×d) , and C ∈ R d×d sym is the right Cauchy-Green tensor The ansatz (2.7) also means that The simplest choice, which is adopted in this paper for avoiding unnecessary technicalities, is that the viscosity σ vi is linear in . C. This is the relevant modeling choice for non-activated dissipative processes with rather moderate rates (in contrast to activated processes like plasticity having nonsmooth potentials that are homogeneous of degree 1 in a small-rate approximation). This linear viscosity leads to a potential which is quadratic in . C, viz.
F). While we will be able to handle general dependence on F, it will be a crucial restriction that Furthermore, the specific dissipation rate can be simply identified in terms of ζ as (2.10) For our choice (2.9), we simply have ξ(F, . C: . In brief, the standard thermodynamical arguments start from the free energy density ψ and the definition of entropy via s = −∂ θ ψ (here H does play no role as it is chosen to be independent of θ ) and the entropy equation with the dissipation rate ξ from (2.10) and the heat flux q. We further use the formula .
. F and the Fourier law formulated in the reference configuration q = −K(F, θ)∇θ, (2.12) which will be specified later in (2.24). Altogether, we arrive at the coupled system div σ vi (∇ y, ∇ . on Q. We complete (2.13) by some boundary conditions. For simplicity, we only consider a mechanically fixed part Γ D time independent undeformed (that is identity) while the whole boundary is thermally exposed with a phenomenological heat-transfer coefficient κ 0: where n is the outward pointing normal vector, and θ is a given external temperature. Moreover, following [10] the surface divergence "div S " in (2.14a) is defined as div S (·) = tr ∇ S (·) , where tr(·) denotes the trace and ∇ S denotes the surface gradient given by ∇ S v = (I − n⊗ n)∇v = ∇v − ∂v ∂ n n. See (2.29) for a short mathematical derivation of the boundary conditions (2.14a) and (2.14c), and [48, pp. 358-359] for the mechanical interpretation in second-order materials.
In order to facilitate the subsequent mathematical analysis, we assume a rather weak thermal coupling through the free energy (together with the coupling through the temperature-dependent viscous dissipation). To distinguish the particular coupling thermo-mechanical term from the purely mechanical one, we consider the explicit ansatz (2. 15) In applications, the internal energy e given by Gibbs' relation is often balanced. Here, we rather use the thermal part of the internal energy w := e − ϕ(F). In view of the ansatz (2.15), we have (2.16) Note that w(F, ·) is the primitive function of the specific heat c v (F, ·) calibrated as w(F, 0) = 0, in accord with the fact that w = e − ϕ(F) = e − ψ(F, 0). The heat-transfer equation (2.13b) simplifies as (2.17) In particular, the purely mechanical stored energy ϕ does not occur in (2.16) and does not influence the heat production and transfer in (2.17). This is crucial because in the first step of the analysis we are not able to control the determinant of ∇ y, but ∂ F ϕ(∇ y, θ) blows up for det ∇ y 0. In contrast, we are able to assume that F → ∂ F φ(F, θ) behaves globally nicely, see (2.30b) and (2.30c).
The energetics of the system (2.13)-(2.14) can be best described by introducing additional energy functionals as follows: Using σ el = ∂ F ϕ + ∂ F φ and integrating in time leads to the relation y dt, (2.20) which will be very useful for obtaining a priori estimates in the sections to follow. Next, we test the heat equation in its simplified form (2.17) together with the boundary conditions (2.14d) by the constant function 1 (that is we merely integrated over Ω) and add the result to (2.20). After major cancellations we obtain the total energy balance: (2.21) In particular, we see that the total energy is conserved up to the work induced by the external loadings or the flux of heat through the boundary. From the entropy equation (2.11), we can read the total entropy balance (the Clausius-Duhem inequality): entropy flux through boundary dS. (2.22) This articulates, in particular, the second law of thermodynamics that the total entropy in the isolated systems (that is here q = 0 on Γ ) is nondecreasing with time provided K = K(∇ y, θ) is positive semidefinite and the dissipation rate is non-negative. It is certainly a very natural modeling choice that Fourier's law is formulated in the actual (also called the deformed) configuration in a simple form, namely the actual heat flux is given by q = −K(θ)∇ z θ, where z = y(x) and θ(z) = θ(y −1 (z)) for z ∈ y(Ω) (2.23) with the heat-conductivity tensor K = K(x, θ) considered as a material parameter possibly dependent on x ∈ Ω. We transform (that is pull back) this Fourier law to the reference configuration via q = (Cof F ) q, because fluxes should be considered as (d−1)-forms. Writing Fourier's law in material coordinates as q(x) = K(x)∇θ a comparison with (2.23) leads to the usual transformation rule for 2nd-order contra-variant tensors, namely if det F > 0, whereas the case det F 0 is considered nonphysical, so K is then not defined. Here we used the standard shorthand notation F − = [F −1 ] = [F ] −1 and also the algebraic formula In what follows, we omit explicit x-dependence for notational simplicity. Let us emphasize that in our formulations ∇θ is not treated as a vector, but a contravariant 1-form. Starting from θ(x) = θ(y(x)) the chain-rule gives ∇θ(x) = ∇ y(x) ∇ Y θ (y(x)). It should be noted that (2.23) is a rather formal argumentation, assuming injectivity of the deformation y and thus existence of y −1 , which is however not guaranteed in our model; anyhow, handling only local non-selfpenetration while ignoring possible global selfpenetration is our modeling approach often accepted in engineering, too, see for example [49, p. 433], [47,Sec. 3.1], and [48, p. 293].
For the isotropic case K(θ ) = κ(θ )I, relation (2.24) can also be written by using the right Cauchy-Green tensor C = F F as K = det(F)κ(θ )C −1 , cf. for example [17,Formula (67)] or [23,Formula (3.19)] for the mass instead of the heat transport. In principle, K in (2.23) itself may also depend on C = F F, which we omitted to emphasize that K in (2.24) will depend on F even if K itself will not.
In what follows, we will use the (standard) notation for the Lebesgue L pspaces and W k, p for Sobolev spaces whose kth distributional derivatives are in L p -spaces and the abbreviation H k = W k,2 . The notation W 1, p D will indicate the closed subspace of W 1, p with zero traces on Γ D and set p = p/( p−1). Thus, for example, (2.25) For the fixed time interval I = [0, T ], we denote by L p (I ; X ) the standard Bochner space of Bochner-measurable mappings I → X with X a Banach space. Also, W k, p (I ; X ) denotes the Banach space of mappings from L p (I ; X ) whose k-th distributional derivative in time is also in L p (I ; X ). The dual space to X will be denoted by X * . Moreover, C w (I ; X ) denotes the Banach space of weakly continuous functions I → X . The scalar product between vectors, matrices, or 3rd-order tensors will be denoted by " · ", ":", or " . . . ", respectively. Finally, in what follows, K denotes a positive, possibly large constant. We consider an initial-value problem, imposing the initial conditions y(0, ·) = y 0 and θ(0, ·) = θ 0 on Ω. (2.26) Having in mind the form (2.17) of the heat equation, we can now state the following definition for a weak solution: if (y, θ) ∈ C w (I ; W 2, p (Ω; R d )) × L 1 (I ; W 1,1 (Ω)) with ∇ . y ∈ L 2 (Q; R d×d ), if min Q det ∇ y > 0 and y| Σ D = identity, and if it satisfies the integral identity for all smooth z : Q → R d with z = 0 on Σ D together with y(0, ·) = y 0 , and if At first sight, it seems that (2.27a) is not suited to apply the test function z = .
y, which is the natural and necessary choice for deriving energy bounds. Obviously, we will not be able to obtain enough control on ∇ 2 . y. However, using the abstract chain rules provided in Section 3.3 this problem can be handled by extending H(y) = Ω H (∇ 2 y) dx to a lower semicontinuous and convex functional on , see the rigorous proof of (5.9) in Step 3 of the proof of Proposition 5.1.
It will be somewhat technical to see that the weak formulation (2.27a) is indeed selective enough, in the sense that for sufficiently smooth solutions one can indeed obtain the classical formulation (2.13) together with the boundary conditions (2.14), cf. also [44,Sect. 2.4.4]. In particular, abbreviating σ = σ vi (∇ y, ∇ . y, θ) + σ el (∇ y, θ), integrating by part once, and using the boundary conditions (2.14a,c) yields We now want to show how the strong form (2.13a) and the associated boundary conditions (2.14a,c) follow from (2.28). For this goal, we apply Green's formula in the opposite direction to remove ∇ in front of the test function z. Using also the orthogonal decomposition of ∇z = ∇ S z+ ∂ ∂ n z⊗ n involving the surface gradient ∇ S z and writing shortly h for h el (∇ 2 y) ∈ R d×d×d , relation (2.28) leads to the identity Using the surface divergence div S and the projection P S : A → A − A n ⊗ n to the tangential part, we obtain the integration by parts formula (cf. [10] or [48, pp. 358-359]) where the surface Γ is now assumed to be sufficiently smooth. Using this with A = h n for the previous relation we find where we have used z = 0 on Σ D = Σ\Σ N . Now, taking z's with a compact support in Q, we obtain the equilibrium (2.13a) in the bulk. Next taking z's with zero traces on Σ but general ∂z ∂ n , we obtain (2.14c). Note that the latter condition implies P S (h n) = h n − h : ( n⊗ n) ⊗ n = h n. Hence, taking finally general z's, we obtain (2.14a), as P S can be dropped because of (2.14c).
Moreover, also note that, from the integral identity (2.27b), one can read w(∇ y(0), θ (0)) = w(∇ y 0 , θ 0 ) from which θ(0) = θ 0 follows when taken the invertibility of w(F, ·) and y(0) = y 0 into account. Now we exploit the decomposition (2.15) of ψ into ϕ and φ, which allows us to impose coercivity assumptions for the purely elastic part ϕ that are independent of those for φ: where GL + (d) denotes the set of matrices in R d×d with positive determinant. The last assumption in (2.30c) means that c v together with c −1 v are bounded, which is a major restriction. However, it allows for a rather simple estimation in Lemma 6.3; for alternative, more general situations dealing with increasing c v (·) we refer to [26,Sec. 8.3].
Assumptions (2.30b,c) make the thermomechanical coupling through φ rather weak in order to allow for a simple handling of the mechanical part independently of the temperature. These restrictive assumptions are needed for our specific and simple way of approximation method rather than for the problem itself. E.g. the assumption in (2.30b) is used to facilitate the estimate (4.12), which allows us to control the difference between Ω φ(∇ y k , θ) dx and Ω φ(∇ y k−1 , θ) dx in terms of M(y k ), M(y k−1 ), and ∇ y k −∇ y k−1 2 L 2 . Moreover, after having derived uniform bounds on |∇ y k | it will be exploited to show that the thermo-coupling stress ∂ F φ is bounded. Finally, (2.30d) and (2.30h) make the stored energy finite at time t = 0.
It will be important that ∂ F φ(F, θ) vanishes for θ = 0 [which follows from (2.15)], so that temperature stays non-negative if θ 0 0 and θ 0, as assumed. We now state our main existence results, which will be proved in the following Sections 4-6. The method will be constructive, avoiding non-constructive Schauder fixed-point arguments, however some non-constructive attributes such as selections of converging subsequences will remain. More specifically, the proof is obtained by first making the a priori estimates for time-discretized solutions in Proposition 4.2, and then deriving an existence result for time-continuous solutions of an ε-regularized problem, see Proposition 5.1. Finally, Proposition 6.4 provides convergence for ε → 0. Theorem 2.2 (Existence of energy-conserving weak solutions). Assume that the conditions in (2.30) hold. The original initial-boundary-value problem (2.13)-(2.14)-(2.26) with K from (2.24) possesses at least one weak solution (y, θ) in the sense of Definition 2.1. In addition, these solutions satisfy ∇θ ∈ L r (Q; R d ) for all 1 r < (d+2)/(d+1), the mechanical energy balance (2.19), and the total energy balance (2.21).
As mentioned in the introduction, a lot of publications are devoted to the simpler isothermal viscoelasticity at large strain, yet, in the multi-dimensional case, they do not satisfy all the necessary physical requirements. It is therefore worthwhile to present a version of our existence result by restricting it to this isothermal case, for which a lot of assumptions are irrelevant or simplify. In particular, (2.15) simplifies as ψ(F, θ) = ϕ(F). Of course, our theory only works because we are using a non-degenerate second-grade material, where the energy contribution H(y) := Ω H (∇ 2 y) dx generates enough regularity to handle the geometric and physical nonlinearities. To the best of the authors knowledge, even the result for isothermal viscoelasticity is new.
A similar regularization approach to isothermal large-strain viscoelasticity was considered in [20], where the H(y) is multiplied with a small parameter that vanishes slower than the loading. Hence, the authors are able to show that their solutions are sufficiently close to the identity which allows them to exploit a simpler Korn's inequality obtained by a perturbation argument. Hence, to the best of the authors' knowledge the following result is the first that allows for truly large strains: Before going into the proof of our main result, we show that our conditions are general enough for a series of nontrivial applications: Example 2.4 (Classical thermomechanical coupling). The classical example of a free energy in thermomechanical coupling is given in the form that is φ(F, θ) involves a term in the product form −a(θ )ϕ 1 (F). For the purely mechanical part we may take the polyconvex energy ϕ(F) = c 1 |F| s + c 2 /(det F) q for det F > 0 and ∞ otherwise. For the thermomechanical coupling we obtain thus to have positivity of the heat capacity c v , we assume a (θ ) 0 and ϕ 1 (F) 0. Moreover, we have Thus, we see that all assumptions in (2.30) can easily be satisfied, for example by choosing a(θ ) = (1+θ) −α with α > 0, which is smooth bounded and convex, and taking any φ 1 ∈ C 2 c (R d×d ).
Example 2.5 (Phase transformation in shape-memory alloys). An interesting example of a free energy ψ occurs in the modeling of austenite-martensite transformation in so-called shape-memory alloys: cf. for example [42] and references therein. Here a denotes the volume fraction of the austenite versus martensite which is supposed to depend only on temperature.
Of course, this is only a rather simplified model.
The heat capacity then reads as To ensure its positivity, ψ 0 has to be strictly concave in such a way that ψ 0 (θ ) K /θ and then inf (F,θ) θa (θ )[ϕ A −ϕ M ](F) + K > 0 has to (and can) be ensured by suitable modeling assumptions.
Remark 2.6 (Thermal expansion). Multiplicative decomposition F = F el F th with the "thermal strain" F th = I/μ(θ ) and the elastic strain F el which enters the elastic part of the stored energy ϕ. This would lead to Unfortunately, (2.33) is inconsistent with the ansatz (2.15) because the contribution ϕ which has been important for our analysis due to uniform coercivity, cannot be identified in (2.33).

A Few Auxiliary Results
In this subsection we provide a series of auxiliary results that are crucial to tackle the difficulties arising from large-strain theory. First, we show how the theory developed by Healey and Krömer [24] allows us to derive a positive lower bound for det ∇ y from the a priori bounds for the elastic energy M(y, θ). This can then be used to establish the validity of the Euler-Lagrange equations and a useful λconvexity result, which is needed for obtaining optimal energy estimates. Second, we provide a version of Korn's inequality from Pompe [41] that allows us to obtain dissipation estimates via D(y, . . Finally, in Section 3.3 we provide abstract chain rules as derived in [37, Sec. 2.2] that allow us to derive energy balances like (2.20) from the corresponding weak equations.

Local Invertibility and Euler-Lagrange Equations
A crucial point in the large-strain theory is the blow-up of the energy density ψ(F, θ) for det F 0. Thus, it is desirable to find a suitable positive lower bound for det ∇ y(t, x). The following theorem is an adaptation of the result in [ Proof. We give the full proof, since our mixed boundary conditions are not covered in [24]. From M(y) C M and the coercivities of ϕ and H we obtain det ∇ y 0 almost everywhere in Ω and the a priori bounds Together with the Dirichlet boundary conditions in Y id we obtain an a priori bound for y in W 2, p (Ω; R d ) and hence also in This proves the first two assertions.
In particular, the function δ: x → det(∇ y(x)) is Hölder continuous as well with Since Ω is a bounded Lipschitz domain, there exist a radius r * > 0 and a constant α * > 0 such that for all x ∈ Ω the sets B r * (x)∩Ω contain an interior Thus, using the Hölder continuity , and |y−x| = r r * for y ∈ C(x), we can estimate as follows: (4) ).
This yields the lower bound δ(x) q r d * exp −C M /c (5) , viz. the third assertion in (3.1).
The last assertion follows via the implicit function theorem.
The most important part of the above result is that the determinant of ∇ y is bounded away from 0. Hence, the function f → ϕ(F), which blows up for det F 0, is evaluated only in a compact subset of GL + (d) ⊂ R d×d such that ∂ F φ and ∂ 2 ϕ exist. Again following [24,Cor. 3.3] we obtain the Gateaux differentiability of M and as well as a useful -semiconvexity result.
For treating Φ el we use the embedding and the limit passage is trivial as the convergence in the integrand is uniform. To derive (3.3) we observe that the convexity of H implies To treat the functional Φ el we apply Theorem 3.1 to y (1) and y (2) , which implies the pointwise bounds Clearly there is a δ > 0 such that We denote by − * the minimum of the smallest eigenvalue of the matrices ∂ 2 F F ϕ(F) where F ∈ R d×d runs through the compact set given by |F| C HK and det F 1/(2C HK ). Hence, assuming ∇ y (2) This establishes the result with (C M ) := max{C HK , 1/δ, * /2}.

A Generalized Korn's Inequality
The following result will be crucial to show that the nonlinear viscosity depending on F = ∇ y really controls the H 1 -norm of the rate .
y. It relies on Neff's generalization [39] of the Korn inequality, in the essential improvement obtained by Pompe [41].
Then, for all K > 1 there exists a constant c K > 0 such that for all F ∈ F K we have Proof. In [41,Thm. 2.3] it is shown that (3.4) holds for each fixed F ∈ F K with c K possibly dependent on F. Let us denote by c(F) > 0 the best possible constants for the given F. By a perturbation argument it is easy to see that the mapping We emphasize that estimate (3.4) is not valid if F is not continuous, see [41,Thm. 4.2]. This shows that the W 2, p -regularity of y is crucial to control the rate of the strain ∇ . y, which is necessary to handle the thermomechanical coupling. The following corollary combines Theorems 3.1 and 3.3, by using the compact embedding W 2, p (Ω; R d ) ⊂ C 1,λ (Ω; R d ).

Chain Rules for Energy Functionals
Abstract chain rules for energy functionals J : X → R ∞ := R∪{∞} on a Banach space concern the question under what conditions for an absolutely continuous curve z: [0, T ] → X the composition t → J (z(t)) is absolutely continuous and satisfies d dt J (z(t)) = Ξ(t), . z(t) for Ξ ∈ ∂J (z(t)), where ∂ denotes a suitable subdifferential. In particular, this implies The case that X is a Hilbert space and J is convex and lower semicontinuous goes back to [13,Lem. 3.3], see also [5,Lemma 4.4].
where ∂J denotes the convex subdifferential, then A first generalization to Banach spaces X with separable dual X * is given in [ By ∂J we denote the Fréchet subdifferential which is defined by Proof. The result follows by the fact that the image of z lies in domJ = { z ∈ X ; J (z) < ∞ } and is compact in X . Hence there is one * < ∞ and one r * > 0 that provides * semiconvexity on B r * (z(t)) for all t ∈ [0, T ]. Thus, the results in the proof of [37,Prop. 2.4] can be applied when choosing ω R ( z, z) = * z−z X there and using the fact that all needed arguments are local and rely only on information of J in a neighborhood of the image of the curve t → z(t).

Time Discretization of a Regularized Problem
Before we construct solutions by a suitable time-discretization, we introduce regularizations in two points. Firstly, we add a linear viscous damping which allows us to obtain simple a priori bounds for the strain rate ∇ . y, because in the first steps of the construction we are not yet in a position to exploit the generalized Korn inequality of Theorem 3.3. Secondly, we modify the heat production induced by the viscous damping, which in the physically correct form leads to an L 1 -source term, that cannot be handled in the first steps of the construction below.
where w is from (2.16) and K from (2.24). This system is defined on Q and is complemented with regularized boundary and initial conditions This system is solved by a time discretization. For this, we consider a constant time step τ > 0 such that T /τ is an integer, leading to an equidistant partition of the considered time interval [0, T ]. (However, varying time-steps can be easily implemented because we will always consider only first-order time differences and one-step formulas.) For the time discretization of the regularized system (4.1)-(4.2) we use the difference notation and define a staggered scheme, where first y k−1 ετ is updated to y k ετ while keeping θ k−1 ετ fixed, and then θ is updated implicitly by updating w k−1 ετ to w k ετ = w(∇ y k ετ , θ k ετ ). More precisely, in the domain Ω we ask for together with the discrete variant of the boundary conditions (4.2) in the form (4.4c) The main advantage is that the boundary-value problem (4.3a), (4.4a), and (4.4b) for y k ετ is the Euler-Lagrange equation of a functional, so that solutions can be obtained by solving the global minimization problem where R is from (2.18e) and where k τ , y = Ω g k τ ·y dx+ Γ N f k τ ·y dS. Clearly, the Euler-Lagrange equation may have more solutions, however for deriving suitable a priori bounds, we will exploit the minimizing properties.
Similarly, the boundary value problem (4.3b) and (4.4c) for θ k ετ , where y k−1 ετ and y k ετ are given, has a variational structure. For this, we define the functions ε we see that W (F, ·) is uniformly convex by assumption (2.30c). Thus, we obtain solutions θ k ετ of (4.3b) and (4.4c) via the minimization problem We emphasize that this staggered scheme is constructed in a very specific way by taking θ = θ k−1 ετ from the previous time step in the mechanics problem for y k ετ , see (4.5). For the construction of θ = θ k ετ from the heat equation we have to use sometimes the explicit (backward) approximations θ k−1 ετ and sometimes the implicit (forward) approximation θ k ετ . Clearly, the former is simpler and it is used in the heat conduction tensor K(∇ y k−1 ετ , θ k−1 ετ ) and in the heat production ξ reg ε . It is tempting to use the explicit choice θ k−1 ετ also in the thermo-mechanical coupling term ∂ F φ(∇ y k ετ , θ):∇δ τ y k ε [= the last term in (4.3b)] as it would simplify the energy balance, see Remark 6.1. However, as this term does not have a sign, we would not be able to guarantee positivity of θ k ετ . Thus we are forced to use the more involved implicit term θ → ∂ F φ C (∇ y k ε , θ):∇δ τ y k ε in (4.7) instead of the simpler, linear choice θ → θ∂ F φ(∇ y k ετ , θ k−1 ετ ):∇δ τ y k ε . This choice may introduce nonconvexity, so that θ k ετ may not be unique. The next result states that we can obtain solutions (y k ετ , θ k ετ ) of (4.3)-(4.4) by solving the minimization problems (4.5) and (4.7), alternatingly. For notational simplicity we have written the minimization problem (4.7) for θ with the constraint θ 0, however, for establishing the Euler-Lagrange (4.3b) and (4.4c) we need to show that non-negativity of θ comes even without imposing the constraint. This will be achieved by minimization over θ ∈ H 1 (Ω) after extending all functionals suitably for θ < 0. Proof. Mechanical step: We first show that the minimization problem in (4.5) has a solution for any θ k−1 ετ ∈ H 1 (Ω) with θ k−1 ετ 0. We cannot rely on that φ is bounded from below, cf. Example 2.4, but we can formally add an x-dependent con- and on K from (2.30c). Since s > 2 is assumed, this possible decay is however dominated by the s-growth of ϕ, cf. (2.30a).
Thus, such formally modified functional in the minimization problem (4.5) is coercive on Y id ⊂ W 2, p (Ω; R d ). By lower semicontinuity in W 2, p (Ω; R d ) we obtain the desired minimizer y k ετ ∈ Y id with M(y k ετ ) < ∞. Hence, Theorem 3.1 shows that the minimizer satisfies det ∇ y(x) δ > 0. As in Proposition 3.2 we conclude that y k ετ satisfies the Euler-Lagrange equation but this gives exactly (4.3a), (4.4a), and (4.4b).
Energy step: We now assume that θ k−1 ετ ∈ H 1 (Ω) and y k−1 ετ , y k ετ ∈ Y id are given with θ k−1 ετ 0 and M(y k−1 ετ ), M(y k ετ ) < ∞. With this, we show that a variant of the minimization problem (4.7) has a minimizer θ k ετ . For this, we extend the function φ, which satisfies φ(F, 0) = 0 by assumption (2.15), continuously by φ(F, θ) = 0 whenever θ < 0. As the functions w, φ C , and W are defined through φ, they all extend continuously differentiable for θ < 0 to the constant value 0. Thus, the integrands in (4.7) are defined for all θ ∈ R and we can minimize over θ ∈ H 1 (Ω), that is without the constraint θ 0.
To show that all global minimizers are non-negative we test the Euler-Lagrange equation by the negative part θ − * := min{θ * , 0} of θ * , which is still an H 1 function: In the first estimate we have used that w k−1 ετ = w(∇ y k−1 ετ , θ k−1 ετ ) εθ k−1 ετ 0, ξ reg ε 0, and θ k ,ε,τ 0 which gives the non-negativity of p 2 , p 4 , and p 7 , while the first and fifth term vanish identically since for θ * > 0 we have θ − * = 0 while for θ * < 0 we have w(F, θ * ) = 0 and ∂ F φ(F, θ * ) = 0 (here we crucially use the implicit structure). Thus, we conclude that θ − * = 0, which is equivalent to θ * 0. Thus, choosing θ k ετ = θ * for any global minimizer of the extended functional we see that it is also a global minimizer of (4.7) and that the Euler-Lagrange equations hold.
The next result provides the basic energy estimates where we will crucially use the carefully chosen semi-implicit scheme defined through the staggered minimization problems (4.5) and (4.7). Here also we will essentially rely on the regularizing viscous term ε . y, as the bounds provided by R cannot be used because of the missing a priori bound for y k ετ in W 2, p (Ω; R d ). Moreover, we will exploit the fact that we have global minimizers in (4.5) rather than arbitrary solutions of the Euler-Lagrange equations (4.3a). This latter argument works because we have neglected inertial terms in the momentum balance (2.27a) and hence in (4.3a). We refer to [26] for cases where inertial effects are treated, but in the isothermal case.
Recalling the notation I = [0, T ], we formulate our first result.
Proof. As y k ετ is a global minimizer, we can insert y = y k−1 ετ as testfunction in (4.5) to obtain the estimate (recall δ τ y k (4.10) The proof will be divided into three steps.
Step 1: Uniform energy bound. Using the decomposition Ψ (y, θ) = M(y) + Φ cpl (y, θ), see (2.18b), we can write (4.10) equivalently as To estimate the last term use the assumption (2.30b) on |∂ F φ(F, θ)| as follows where ρ > 0 is arbitrary. Choosing ρ = ε/(4τ ) and F j = ∇y we can insert this into the estimate (4.11). Moreover we can use R 0 and k τ , δ τ y k Using the coercivity assumption (2.30b) for φ, the second-last term can be estimated by M again and setting m k := M(y k ετ ) we obtain the recursive estimate with c ε = 2·3 s K 2 /ε and c ε = 2c 2 P /ε. In a first step we neglect the last term on the left-hand side and obtain We now restrict τ > 0 via τ < 1/(2 c ε ) by choosing K ε 2 c ε , so we can iterate the above estimate. With (2.30h) we have m 0 := Ψ (y 0 , θ 0 ) < ∞ and a simple induction yields the discrete Gronwall-type estimate (with Using Theorem 3.1 we obtain the desired uniform upper bound in (4.9a) for the interpolant y ετ : , as well as the lower bound (4.9b) for the determinant.
Step 2: Dissipation bound. We return to (4.13) and add all estimates from k = 1 to This provides the uniform bound for y ετ in H 1 (I ; H 1 (Ω; R d )), and (4.9a) is established.
Step 3: Temperature bounds. Testing the Euler-Lagrange equations (4.3b) and (4.4c) by w k ετ yields the identity Recalling c v (F, θ) = ∂ θ w(F, θ) we obtain the chain rule Moreover, we have the elementary estimate 1 2 , and θw = θ w(F, θ) 0 by the definition of w. Using additionally c v (F, θ) = −θ∂ 2 θθ φ(F, θ) ε, see (2.30c), the above identity (4.15) leads to Using the uniform bounds for ∇ y ετ and det ∇ y ετ from Step 1, the assumptions on K in (2.30f), as well as formula (2.24) we find a κ ε such that Fθ φ, the assumptions (2.30b) and (2.30c) together with the uniform L ∞ bound for ∇ y ετ we find ∂ F w(∇ y k ετ , θ k ετ ) L ∞ c ε . Realizing also that we have ∇ 2 y k ετ already estimated in L p (Ω; R d×d×d ) with p 2 we obtain b k ετ L 2 C ε . For the right-hand side h k ετ of (4.15) we have where we again used the L ∞ bounds for ∇ y k ετ . Finally, by definition we have θ ,ε ∈ [0, 1/ε], and (2.31) allows us to estimate w by θ , which yields the boundary estimate Based on the above estimates and introducing the abbreviations we can estimate the right-hand side in (4.17) via where α > 0 is arbitrary. Estimating the last term on the left-hand side in (4.17) from below by ε 2 k /κ ε we may choose α = ε/(2κ ε c ε ). After multiplying (4.17) by 2τ we obtain Arguing as in Steps 1 and 2 for (4.13) and using γ 2 Thus, we find (4.9c) for θ ετ , and estimate (4.9d) follows by using (4.16) once again. The uniform estimate for the piecewise affine interpolant w ετ in the spaces C(I ; L 2 (Ω)) ∩ L 2 ([τ, T ]; H 1 (Ω)) follows from the previous estimates for w ετ . Finally, we note that the time derivative of the interpolant w ετ is equal to δ τ w k ε on the intervals ](k−1)τ, kτ [. We now use the Euler-Lagrange equations (4.3b) and (4.4c), which provides for δ τ w k Squaring and summation over k = 1, . . . , N τ gives the remaining uniform bound in (4.9e) for .

Remark 4.3 (Full space-time discretization).
To be useful for an implementable algorithm, our time discretization should be combined with a spatial discretization. A suitable space discretization itself (leading to the Faedo-Galerkin method) was introduced in [26, Sect. 9.3] even without requiring the ε-regularization (4.1). This suggests that the space-time discretization might work with ε = 0 under a suitable stability criterion that the time step is sufficiently small with respect to the mesh size used for the space discretization. Or, vice versa, our analysis suggests that the ε-regularization makes the stability criterion obsolete, assuming that both time step and mesh size are sufficiently small with respect to ε. However, our focus does not lie in such numerical aspects, and we leave these questions for further research.

The Limit
Using the above a priori estimates for the interpolants we will be able to extract convergent subsequences. First we will observe that the three different types of interpolants have to converge to the same limit. Next we want to pass to the limit in the discretized weak forms of the momentum balance and the heat equation. While most terms can be handled by compactness arguments or weak-convergence methods, there is one term that needs special attention, namely the heat-source term ξ reg ε that is quadratic in ∇ . y ε . Thus, it will be a crucial step to show strong convergence of . y ετ in L 2 (I ; H 1 (Ω)), which can be done by passing to the limit in a suitable discretized version of the mechanical energy balance (2.20). In this argument we will use the -convexity derived in Proposition 3.2 to relate the mechanical energies M(y k−1 ετ ) and M(y k ετ ). With the definition (4.8) for the three types of interpolants, we see that the following discretized version (5.1) of the momentum balance and heat equations (4.1) and (4.2) holds for the discrete solutions constructed in Proposition 4.1: − div σ vi (∇ y ετ , ∇ . y ετ , θ ετ ) + ε∇ .
Here it is essential that we have to use all three types of interpolants, for example y ετ , y ετ , and y ετ . In particular, we emphasize that t → w ετ (t) is the piecewise affine interpolant of {w k ετ } k=0,...,N τ , which does not coincide with t → w(∇ y ετ (t), θ ετ (t)) except at the nodal points t = kτ .
Moreover, any couple (y ε , θ ε ) obtained by this way is a weak solution to the regularized initial-boundary-value problem (4.1)-(4.2) in the sense of Definition 2.1 written for (y ε , θ ε ) with ε∇θ ε : ∇z added in the first integral in (2.27a). Moreover, it satisfies the corresponding mechanical energy-dissipation balance, see (5.9) below.
Proof. The proof consists of five steps.
Step 2: Convergence in the mechanical equation. Now the convergence in the discretized momentum balance (5.1a) can be done by the above weak convergences (5.3) because σ vi is linear in terms of . F and by Minty's trick for the monotone operator induced by h el = H . For a reflexive Banach space X and a hemicontinuous, monotone operator H : X → X * Minty's trick means the implication We apply this for H defined by H(y), z = Q h el (∇ 2 y(t, x)) where X = W 2, p (Q). Clearly, H is hemi-continuous and monotone. Choosing because we can pass to the limit τ in all four terms separately. For the first term we apply the lower semicontinuity result [21, Thm. 7.5] twice, namely for the integrands f ± (x, (F, θ), G) = ±σ vi (F, G, θ):∇z(x) which both are convex in G.
The limit passage in the second term is simple weak convergence, and the fourth term converges because of τ → in L 2 I ; H 1 D (Ω) * . In the third term we exploit ∇ y ετ ∈ F(K ε ) := F ∈ R d×d ; |F| K ε , det F 1/K ε (see (4.9a) and (4.9b) from Proposition 4.2), such that using (2.30a) and (2.30b) the θ) is continuous and bounded on F(K ε ) × R + . Hence, with (5.4) and Lebesgue's dominated convergence theorem we obtain the desired convergence.
To use Minty's trick (5.8) we still need to check b τ , y ετ → b, y ε . However, we have shown above that b τ is bounded (and hence weakly converging to b) in L 2 I ; H 1 D (Ω) * and y ετ → y ε in L 2 I ; H 1 D (Ω) strongly by (5.4a), thus the result follows immediately. Hence, we conclude H(y ε ) = b, which is nothing else but the regularized momentum balance (4.1a), (4.2a), and (4.2b).
Step 3: Balance of mechanical energy. For the limit passage in the heat equation we need strong L 2 -convergence of ∇ . y R(y ε , . y ε , θ ε ) + ε∇ .
Step 4: Strong convergence of strain rates. The next step is now to derive a similar mechanical energy balance for the time-discretized solutions, which is better than the previously used estimate (4.11). Passing to the limit τ → 0 from the latter estimate we would arrive at an estimate like (5.9), but with 2R and ε replaced by R and ε/2, respectively. To improve the discrete estimate (4.11) used in Proposition 4.2 we can exploit the a priori estimates M(y k ετ ) K ε , which allow us to use the geodesic -convexity result in Proposition 3.2. Instead of using the minimization property of y k ετ in (4.5) we test the Euler-Lagrange equation (4.3a) with boundary conditions (4.4a) and (4.4b) by y k ετ −y k−1 ετ to obtain where we have the correct factors 2R and ε. To recover the energy values M(y j ετ ) we now eliminate the term involving DM using the -convexity estimate (3.3) with y (1) = y k ετ and y (2) = y k−1 ετ , which yields We now sum this inequality over k = 1, , . . . , N τ and using the interpolants we obtain the integral estimate ετ . Using lower semicontinuity arguments (use [21,Thm. 7.5] once again for I (2) ετ ) we find Thus, passing to the liminf on the left-hand side and to the limit on the right-hand side in (5.10) and comparing with (5.9) we obtain Together with (5.11) we conclude that we must have equality in all three cases after " ⇒". However, ∇ .
y ε in L 2 (Q; R d×d ) and imply the desired strong convergence ∇ . y ετ → ∇ .
Step 5: Limit in the heat equation. We first pass to the limit τ → 0 in the constitutive relation (5.1b), namely w ετ = w(∇ y ετ , θ ετ ). The left-hand side converges to w ε by (5.7), while the right-hand side converges to w(∇ y ε , θ ε ) by the continuity of w, the bound (2.31) and the convergences (5.4). Thus, w ε = w(∇ y ε , θ ε ) is established, that is (4.1c) holds. We write the heat equation (5.1b) with boundary conditions (5.2c) in the weak form y ετ z dx dt (5.12) for all z ∈ L ∞ (I ; H 1 (Ω)). While we only have the weak convergences . Thus, passing to the limit τ → 0 in (5.12) leads exactly to the weak form to the regularized heat equation (4.1b) with boundary condition (4.2c). This concludes the proof of Proposition 5.1.

Limit Passage
In this final part of the proof of Theorem 2.2 we have to pass to the limit with the regularization parameter ε → 0. As we are already in the time-continuous setting we are now able to make the formally derived total energy balance (2.21) for E rigorous for all ε > 0. From this we will be able to derive a priori bounds for (y ε , θ ε ) that are independent of ε. Remark 6.1 (Missing discrete estimate for the total energy). The derivation of the total energy balance is achieved by testing the momentum balance by .
y and the heat equation by the constant function 1. The corresponding step on the time-discrete level would be the test (4.3a) by δ τ y k and (4.3b) by 1. We would be able to use the desirable cancellation of the dissipation, namely ξ reg ε − ξ 0; however for the coupling terms which arise from (4.3a) and (4.3b) respectively, we do not have any way to estimate the first against the second. Recall that we were forced to use the explicit/forward value θ k ετ to maintain positivity of the temperature.
To exploit the balance of the total energy we have to strengthen the assumption on the loading (t), that is the functions g, and f , in (2.30g), namely , which is what we will only need.
The new ε-independent estimates on ∇ . y ε in L 2 (Q) will be obtained by exploiting Pompe's generalization of Korn's inequality (cf. [41]) as prepared in Theorem 3.3 above. det(∇ y ε ) > 0 on Q and the following estimates hold with K independent of ε > 0: Proof. We proceed in two steps that are close to estimates we have done in the time-discrete setting.
Step 1: Estimate for E(y ε , θ ε ) Using the derived regularity for the solution (y ε , θ ε ) we see that a suitable variant of the total energy balance (2.21) holds. To be specific, we start from (5.9), which is also valid for arbitrary t ∈ ]0, T ] in place of T , and add the time-integrated version of (4.1b) tested with the constant function z ≡ 1.
The important point here is the cancellation of the term ∂ F φ: ∇ . y ε and that the difference of the dissipation integrals has a sign.
Defining the auxiliary variable E ε (t) := E(y ε (t), θ ε (t)) − (t), y ε (t) and using 0 θ ε θ and θ ε 0 gives where we have integrated by parts the power of the external loadings, which was possible by the strengthened assumption (6.1).
With E M H and the coercivity of H we have y H 1 c 1 + c 2 E(y, θ) and obtain with a, b 0 and a, b ∈ L 1 (0, T ), which follows from (6.1) for and (2.30i) for θ . With Hence, (6.2c) is established, whereas (6.2a) and (6.2b) follow by applying Theorem 3.1.
Step 2: Estimate for the strain rate ∇ . y ε We return to the mechanical energy balance (5.9) on the interval I = [0, T ]. We recall that the dissipation function ξ(F, . F, θ) is assumed to control the symmetric part of F . F only, namely Using our a priori bounds on M(y ε (t)), we can apply the generalized Korn inequality prepared in Corollary 3.4 with y = y ε (t, ·) and v = .
From this, we derive an a priori bound on ∇w ε by setting K ε = K(∇ y ε , θ ε ) and estimate it as in (4.18) (see Step 3 of the proof of Proposition 4.2) by where κ is now independent of ε because of the ε-independent bound in (6.2a) and (6.2b). Moreover, ∇w ε and ∇θ ε are related by Subtracting 1 2K κ |∇w ε | 2 , multiplying by χ (w ε ) ∈ [0, 1], and integrating over Q we can employ (6.4) and arrive at where the last integrand is bounded by (6.2a) using p 2.
We are now in the position to pass to the limit ε → 0 in the regularized system (4.1)-(4.2), and thus conclude the proof of our main existence result presented in Theorem 2.2. The approach is close to the convergence result presented in Proposition 5.1: first we extract converging subsequences and then pass to the limit in the mechanical momentum balance. This also provides the necessary strong convergence of the strain rates that is needed to eventually pass to the limit in the heat equation. Proposition 6.4 (Convergence for ε → 0). Let again (2.30) and (6.1) hold. Then, considering the limit ε → 0, there are functions y and θ and a subsequence (y ε , θ ε ) (not relabeled) of weak solutions to the regularized system (4.1)-(4.2) obtained in Proposition 5.1 such that it holds that y ε → y weakly* in L ∞ (I ; W 2, p (Ω; R d )) ∩ H 1 (I ; L 2 (Ω; R d )) and (6.7a) θ ε → θ weakly in L r (I ; W 1,r (Ω)) for all 1 r < (d+2)/(d+1). (6.7b) Moreover, every couple (y, θ) obtained in such a way is a weak solution, according Definition 2.1, of the boundary-value problem (2.13)-(2.14) satisfying the initial conditions (2.26).
Proof. The proof follows the lines of the proof of Proposition 5.1, so we do not repeat all details of the arguments.
Step 2: Convergence in the mechanical equation. The limit passage in the momentum balance (4.1a)-(4.2) works as before, again using the Minty trick (5.8). Of course, the additional regularizing viscosity term ε∇ .
Step 3: Balance of mechanical energy. As in the proof of Proposition 5.1 we derive from the property that the limit couple (y, θ) solves the mechanical equation that the following mechanical energy relation holds (again exploiting the chain rule in Proposition 3. Step 4: Strong convergence of the symmetric strain rates. We can pass to the limit ε → 0 in the mechanical energy relation (5.9). Comparing the result with (6.9) we obtain lim ε→0 Q ξ(∇ y ε , ∇ . F F, θ). From the pointwise convergence θ ε → θ , the uniform convergence F ε := ∇ y ε → F = ∇ y, and the weak convergence With the coercive and quadratic structure of ζ assumed in (2.30e) we proceed as follows: where C ε = F ε F ε . We see that the first term converges by (6.10), while the second term converges by the weak convergence V ε V and the strong convergence D(C ε , θ ε )V → D(C, θ)V (as D is bounded and the arguments converge pointwise). Similarly, δ(ε) → 0 by Lebesgue's dominated convergence theorem, and thus we conclude the strong convergence V ε −V L 2 (Q;R d×d ) → 0.
As the majorants g ε v L ∞ (Q) converge to g v L ∞ (Q) in L 1 (Q) the generalized dominated convergence theorem implies convergence of the second term in (6.11).
In the third term we have weak convergence of ∇ . y ε and strong convergence of v∂ F φ(∇ y ε , θ ε ). Similarly, the remaining four terms converge to the desired limits. Thus, we have shown that (y, θ) satisfy (2.27b), which finishes the proof of Proposition 6.4. Remark 6.5 (Strong convergence of y ετ and y ε ). Strengthening the monotonicity of h el (·), implied by the convexity assumed in (2.30d), to the strict monotonicity we can use the argumentation after (5.11) to show y ετ (t) → y ε (t) strongly in W 2, p (Ω; R d ) for all t ∈ [0, T ]. Similarly, in Proposition 6.4 one can show y ε (t) → y(t) strongly in W 2, p (Ω; R d ). Together with the L ∞ -estimate (4.9a), we can also strengthen the weak* convergence (5.3a) in L ∞ (I ; W 2, p (Ω; R d )) to a strong convergence in L q (I ; W 2, p (Ω; R d )) for all q ∈ [1, ∞[. The same applies to (6.7a).
Remark 6.6 (Dynamical problems). Introducing the kinetic energy 1 2 | . y| 2 with a mass density = (x) > 0 leads to an inertial force .. y in the momentum equation (2.13a), which would make the nonlinear problem hyperbolic. It is generally recognized as analytically very troublesome. Here, it would work for isothermal situations like in Corollary 2.3 if we were able to work with weak convergence, that is H needs to be quadratic ( p = 2). Staying with H depending on the second gradient ∇ 2 y we would be forced to give up the determinant constraint det ∇ y > 0, which is indeed possible if heat conduction is not considered. Alternatively, one may take H quadratic but coercive on the Hilbert space H s (Ω) with s > 1 + d/2, such that H s (Ω) still embeds into C 1,α for some α > 0, cf. also [26,Sect. 9.3]. In the non-isothermal situation, it seems difficult to ensure that the acceleration .. y ∈ L 2 (I ; H 1+κ (Ω; R d×d ) stays in duality with the velocity . y. Obtaining enough regularity is difficult and the higher-order viscosity is inevitably very nonlinear to comply with frame-indifference, while the corresponding generalization of Korn's inequality does not seem to be available. Remark 6.7 (Other transport processes: flow in porous media). Beside heat transport, one can also consider other transport processes in a similar way. The transport coefficients can be pulled back as in (2.24). For example, considering mass transport for a concentration c one has to make the free energy ψ also c-dependent and to augment it by a capillarity-like gradient term 1 2 κ|∇c| 2 . The dissipation potential R will then be augmented by the nonlocal term 1 2 M(∇ y, c) 1/2 ∇ −1
y + ∇μ · M(x, ∇ y, c)∇μ (6.12c) with σ vi as in (2.13a), c v (F, c, θ) = −θ∂ 2 θθ ψ(F, c, θ), and ξ from (2.10). In (6.12b), the variable μ is called the chemical potential that is thermodynamically conjugate to c. One can also augment the model by some inelastic (plastic or creep-type) strain like in [45], where also the inertial forces are included, whereas viscosity is ignored and the restriction to small elastic strains is imposed.