A local existence result for system of viscoelasticity with physical viscosity

We prove the local in time existence of the classical solutions to the system of equations of isothermal viscoelasticity with clamped boundary conditions. We deal with a general form of viscous stress tensor $\mathcal{Z}(F,\dot F)$, assuming a Korn-type condition on its derivative $D_{\dot F}\mathcal{Z}(F, \dot F)$. This condition is compatible with the balance of angular momentum, frame invariance and the Claussius-Duhem inequality. We give examples of linear and nonlinear (in $\dot F$) tensors $\mathcal{Z}$ satisfying these required conditions.

Here, ξ : Ω × R + −→ R n denotes the deformation of a reference configuration Ω ⊂ R n which models a viscoelastic body with constant temperature and density. A typical point in Ω is denoted by X, and the deformation gradient, the velocity and velocity gradient are given as: In (1.1) the operator div stands for the spacial divergence of an appropriate field. We use the convention that the divergence of a matrix field is taken row-wise. In what follows, we shall also use the matrix norm |F | = (tr(F T F )) 1/2 , which is induced by the inner product: F 1 : F 2 = tr(F T 1 F 2 ). To avoid notational confusion, we will often write F 1 : F 2 instead of F 1 : F 2 .
1.1. The elastic energy density W . The mapping DW : R n×n −→ R n×n in (1.1) is the Piola-Kirchhoff stress tensor which, in agreement with the second law of thermodynamics [8], is expressed as the derivative of an elastic energy density W : R n×n −→ R + .
The principles of material frame invariance, material consistency, and normalisation impose the following conditions on W , valid for all F ∈ R n×n and all proper rotations R ∈ SO(n): (1.5) (i) W (RF ) = W (F ), (ii) W (F ) → +∞ as det F → 0, (iii) W (Id) = 0. Examples of W satisfying the above conditions are: where q > 1 and W is intended to be +∞ if det F ≤ 0 [22]. Another case-study example, satisfying (i) and (iii) is: We will assume that W is smooth in a neighborhood of SO(n). Since div(DW (∇ξ)) is a lower order term in (1.1), it follows that other properties of W play actually no role in the proof of our main Theorem 1.1 and 1.2. We hence remark that the same results are valid when div(DW (∇ξ)) is replaced by div(DW ((∇ξ)A(X) −1 )). Such term corresponds to the so-called non-Euclidean elasticity, where the deformation ξ of the reference configuration strives to achieve a prescribed Riemannian metric g = A T A on Ω. This model pertains to the description of prestrained materials and morphogenesis of growing tissues [19,18].
1.2. The viscous stress tensor Z. The viscous stress tensor is given by the mapping Z : R n×n × R n×n −→ R n×n , depending on the deformation gradient F and the velocity gradient Q. It should be compatible with the following principles of continuum mechanics: balance of angular momentum, frame invariance, and the Claussius-Duhem inequality [8]. That is, for every F, Q ∈ R 3×3 with det F > 0, we require that: Examples of Z satisfying the above are: (1.7) We note that in the case of Z ′ 0 , the related Cauchy stress tensor T ′ 0 = 2(detF ) −1 Z 2 F T = 2sym(QF −1 ) is the Lagrangean version of the stress tensor 2sym∇v written in the Eulerian coordinates. For incompressible fluids 2div(sym∇v) = ∆v, giving the usual parabolic viscous regularization of the fluid dynamics evolutionary system.
1.3. The main results. Our main assumption implying the dissipative properties of (1.1) will be expressed in terms of the following condition on a (constant coefficient) linear operator M : R n×n → R n×n : Note that (1.8) is a Korn-type estimate, reducing to the classical Korn inequality for M(F ) = symF and γ = 2 [16]. Naturally, (1.8) is equivalent to (2.1) which is the same estimate valid for all ζ ∈ W 2 1 (U, R n ) with ζ |∂U = 0, on any fixed open, bounded U ⊂ R n . It can be shown, via Fourier transform (see Lemma 2.2), that (1.8) is also equivalent to the strict positive definiteness of M when restricted to the space of rank-one matrices Q = a ⊗ b: We point out that the above condition resembles, naturally, the local well-posedness criterion for the inviscid elasticity system [15], where the validity of (1.9) for M = DW (∇ξ 0 ) is equivalent to the hyperbolicity of the first-order system (1.1) with Z = 0.
With the same techniques of proof of Theorem 1.1, one can show that: Theorem 1.2. Let S be the solution operator of the problem (1.1) -(1.4) as described in Theorem 1.1, given by: . Then S is continuous.
We omit the proof and refer instead to standard texts [1,17,20,21], or to an application of the same methods in the more current context as in Theorem 1.2 [9] .
1.4. Relation to previous works. The dynamical viscoelasticity (1.1) has been the subject of vast studies in the last decades. For Z(F, Q) = Q conflicting with the frame invariance (1.6) (ii), various results on existence, asymptotics and stability have been obtained in [2,23,24,12]. For dimension n = 1, existence of solutions to (1.1) has been shown in [7,4] for Z depending nonlinearly on Q.
Existence and stability of viscoelastic shock profiles for a large class of models originating from (1.1) has been studied, among others, in [3,5].
Existence of Young measure solutions to system (1.1) was shown in [10], without any additional assumptions on Z, but with condition (1.6) (iii) strengthened to the uniform dissipativity i.e: Z(F, Q) ≥ γ|Q| 2 . These measure-valued solutions were shown to be the unique classical weak solutions under the extra monotonicity assumption: see also [25] for a treatment of slightly more general type of PDEs under the same condition. As noted in [10], (1.12) is incompatible with the balance of angular momentum (1.6) (i). In particular, (1.12) is not satisfied by any of the examples in (1.7), even Z 0 , Z ′ 0 , Z ′′ 0 which enjoy condition (1.11) for any invertible F = ∇ξ 0 (X) and any Q = ∇ξ 1 (X).
From the theory of PDEs viewpoint, our present result is a rather straightforward application of the theory of nonlinear (quasilinear) parabolic systems. Namely, we apply the maximal regularity estimates to control the nonlinearities of the system (1.1). We choose the L p -framework in order to avoid technical difficulties, but a similar results and estimates are expected in the Besov spaces framework [9]. In a sense, our result is hence a consequence of the classical works of Ladyzhenskaya, Solonnikov and Uralceva [17], which has been further developed in [1,11,20], and which is a powerful tool in the study of the parabolic-elliptic systems.
1.5. Notation. By L p (Ω) we denote the space of functions integrable with respect to the Lebesgue measure, with p-th power. By W k,l p (Ω × (0, T )) for k, l ∈ N we denote the anisotropic Sobolev space defined by the norm : where ∇ k is the k-th space derivative and ∂ t is the time derivative. The isotropic version is given by: . For further details we refer to [6]. Vladimir Sverak for helpful consultations. M.L. was partially supported by the NSF grant DMS-0846996, and both authors were partially supported by the Polish MN grant N N201 547438.

The constant coefficient problem
The following auxiliary result will be needed in the proof of Theorem 1.1: Lemma 2.1. Assume that M : R n×n → R n×n is a linear map satisfying the Korn-type inequality: Then the solution to: in U admits the following maximal regularity estimate: , where the dependence of C on U is uniform for any family of domains which are uniformly bilipschitz homeomorphic to each other after appropriate dilations.
Towards a proof of Lemma 2.1, note first that for M = Id, i.e. when (2.1) holds trivially, (2.3) is a classical maximal regularity parabolic estimate for the heat equation. When M(F ) = symF , i.e. when (2.1) reduces to Korn's inequality, the proof of (2.3) is also immediate. For, take div of the equation in (2.2), and note that div T div(sym∇ζ) = div 1 2 ∆ζ + 1 2 ∇divζ = 1 2 (div∆ζ + 1 2 ∆divζ) = ∆divζ so that: By the maximal regularity estimate for the heat equation: Now, (2.2) can be written as: We hence obtain: In the general case, Lemma 2.1 follows from the maximal regularity theory developed for parabolic initial-boundary value problems in [11]. Under the ellipticity condition (b) on page 98 in there (see also Definition 5.1), the estimate (2.3) is a consequence of Theorem 7.11. We now prove that condition (2.1) implies that the constant coefficient operator −div (M∇ζ) has its spectrum contained in the proper sector of the complex plane, which immediately gives ellipticity in the sense of [11].

2.
To prove (2.5), consider the eigenvalue problem: which after passing to the Fourier variable k ∈ R n becomes: Upon writing λ = σ|k| 2 , the problem (2.7) is equivalent to locating the eigenvalues σ of the family of linear operators Recalling (1.9) we see that each M k is strictly positive definite: Consequently, spectrum of every M k satisfies Re σ > 0. By continuity with respect to k which varies in the compact set |k| = 1, we obtain the inclusion (2.5).
Finally, we have the following: If we additionally assume that det sym(Q 0 F −1 0 ) = 0 then we also have: The proof of Lemma 2.3 will be given in section 5. We now remark that in the proof of the main Theorem 1.1, Lemma 2.1 will be used to the operators M = M X = D Q Z(F 0 , Q 0 ), at finitely many spacial points X ∈ Ω, where F 0 = ∇ξ 0 (X) and Q 0 = ∇ξ 1 (X). It is clear that when the initial data ξ 0 , ξ 1 with regularity (1.10) satisfy det ∇ξ > 0 (or the two conditions det ∇ξ > 0 and det sym(∇ξ 1 (∇ξ 0 ) −1 ) = 0 whenever required) then the constants γ in Lemma 2.3 have a common upper and lower bounds, independent of X. Therefore, Lemma 2.1 and the estimate (2.3) may be used with a uniform constant C p,U , also independent of X.
whereξ is as in (3.1). Then, there exists T 00 < T 0 and a constant C, both depending only on ξ 0 and ξ 1 (and, naturally, on Ω and p), such that for every T < T 00 we have: In particular: Before we give the proof of the lemma, we gather below some standard inequalities that will be frequently used for different functions: u defined on Ω × (0, T ), and w defined on Ω. We always assume that T < 1.
. The inequality (3.5) is the usual elliptic estimate [14], and (3.6) is the parabolic estimate from [6]. The Morrey embedding gives (3.7) for p > n + 2 [14], while (3.8) follows from the embedding ∇W 2,1 p (Ω × (0, T )) ⊂ L ∞ (Ω × (0, T )), also valid for p > n + 2 [17]. We stress that the constants C in all the above bounds are universal, i.e. they are independent of T . Additionally, the dependence of C in (3.8) on U ⊂ Ω is uniform for any family of domains which are uniformly bilipschitz homeomorphic to each other after appropriate dilations.
In all the above inequalities (3.10) -(3.13), we write Θ = Θ(T ). The constant C depends only on the initial data of the problem ξ 0 , ξ 1 (in addition to its dependence on Ω and p).

2.
We will now work with the localizations of the system (3.15). Let {B k } N k=1 be a covering of Ω by a finite number N = N(r) of balls B k = B(X k , r) with centers X k ∈ Ω and radius r < 1. This family of coverings (parametrized by r) should such that all the sets 2B k ∩ Ω are uniformly bilipschitz homeomorphic to each other after appropriate dilations and that the covering numbers of {2B k ∩ Ω} k are independent of r.
Taking now T 00 so small that, in addition to other requirements imposed in the course of the proof, ǫ = T 1/p + D satisfies (3.28), we obtain that for every T ∈ [0, T 00 ) the quantity Θ(T ) must stay below Θ 0 , in virtue of continuity of the function T → Θ(T ) and Θ(0) = 0. This ends the proof of (3.4) and of Lemma 3.1.

3.
A modification of arguments in section 3 implies that the weak solution ξ is actually regular in the class determined by (4.3), i.e: Moreover, for every small ǫ > 0:

4.
In now suffices to show that the map T is a contraction in some ballB ǫ ⊂ E Ω,T . This is done by applying methods of (3) to the system: where T (ξ i −ξ) = ξ i −ξ. For ǫ > 0 sufficiently small it follows that: which completes the proof.
Proof. Consider a covering {B k } N k=1 of Ω by a finite number N = N(r) of balls B k = B(X k , r) with centers X k ∈ Ω and radius r > 0. This family of coverings (parametrized by r) should be such that their covering numbers are uniform in r. Let {π k } N k=1 be a partition of unity subject to {B k }.

5.
A proof of Lemma 2.3

3.
To prove (iii) -(v), observe that: where we denoted: A = sym(Q 0 F −1 0 ), B = sym(QF −1 0 ). Since the matrix 2m j=0 A j BA 2m−j is symmetric, it follows that: Let ζ be a test function as in Lemma 2.1. By calculations similar to (5.1) we get: proving (iii). To prove (iv), we compute: Therefore, by calculations similar to (5.1): Finally, in order to prove (v) we derive: which, in the same manner as in (5.2) yields: The proof of Lemma 2.3 is done.