On existence of a classical solution to a generalized ...ncmm.karlin.mff.cuni.cz/preprints/11261134953pr16.pdf · ON EXISTENCE OF A CLASSICAL SOLUTION TO A GENERALIZED KELVIN-VOIGT

Necas Center for Mathematical Modeling

On existence of a classical solution toa generalized Kelvin-Voigt model

M. Bulıcek, P. Kaplicky and M. Steinhauer

Preprint no. 2011-016

Research Team 1Mathematical Institute of the Charles University

Sokolovska 83, 186 75 Praha 8http://ncmm.karlin.mff.cuni.cz/

ON EXISTENCE OF A CLASSICAL SOLUTION TO

A GENERALIZED KELVIN-VOIGT MODEL

MIROSLAV BULICEK, PETR KAPLICKY, AND MARK STEINHAUER

Abstract. We consider a two-dimensional generalized Kelvin-Voigt model

describing the motion of a compressible viscoelastic body. We establish theexistence of a unique classical solution to such a model in the spatially periodic

setting. The proof is based on Meyers’ higher integrability estimates that

guarantee the Holder continuity of the gradient of displacement.

1. Introduction

In this paper we focus on qualitative properties of a solution to a generalizedKelvin-Voigt model that describes the motion of a two-dimensional compressibleviscoelastic body. Hence, assuming that the body occupies a domain Ω := (0, 1)2

and that T > 0 is the length of time interest, such a model is described by thefollowing system of equations:

(1.1)

%0utt − divTTT = %0f in Q,

u(0, ·) = u0(·) in Ω,

ut(0, ·) = v0(·) in Ω,

where Q := (0, T )×Ω. Here, %0 : Ω→ R+ is the given density of the body, assumedto be time independent, f : Q → R2 is the given density of external body forces,u : Q → R2 denotes an unknown displacement field and TTT : Q → R2×2 standsfor the Cauchy stress tensor. The initial displacement is denoted by u0 : Ω → R2

and the initial velocity of the body is v0 : Ω → R2. In the context of continuummechanics, (1.1)1 represents the balance of linear momentum (here we already usedthat we consider a simplified model, so that the density is assumed to be timeindependent). Note that using also the balance of angular momentum, the naturalrequirement for non-polar materials is

TTT = TTTT in Q.

Concerning the boundary condition, we restrict ourselves to periodic (with respectto Ω) boundary conditions that require some normalization condition (in order to

2000 Mathematics Subject Classification. 35B65,35Q74,74D10.Key words and phrases. Kelvin-Voigt model; Regularity; Classical Solution; Large-data and

Long-time.M. Bulıcek is supported by the Necas Center for Mathematical Modeling, project LC06052,

financed by MSMT. P. Kaplicky is supported by Grant 201/09/0917 of CSF, and also partially

by the research project MSM 0021620839 financed by MSMT. M. Steinhauer acknowledge thesupport of the Necas Center for Mathematical Modeling, project LC06052, and thanks the Center

for its hospitality.

1

2 M. BULICEK, P. KAPLICKY, AND M. STEINHAUER

guarantee uniqueness of a solution). For simplicity we choose the simplest one

(1.2)

∫Ω

ρ0(x)u(t, x) dx =

∫Ω

ρ0(x)ut(t, x) dx = 0 for all t ∈ (0, T ).

A direct consequence of (1.2) is that we need to assume a compatibility conditionon the data, namely, for all t ∈ (0, T ) we need that

(1.3)

∫Ω

ρ0(x)f(t, x) dx =

∫Ω

ρ0(x)v0(x) dx =

∫Ω

ρ0(x)u0(x) dx = 0.

Although we use the simplest possible boundary condition we believe that ourresult can be adopted to a more general setting with more reasonable physicalboundary data.

Finally, we discuss the constitutive relation for the Cauchy stress TTT we are in-terested in. We assume that TTT is given as a sum of a viscous and an elastic part,i.e.

(1.4) TTT := TTTv +TTTe.

In the classical model (see Kelvin [19], Voigt [21]), such a form of TTT is obtained bylinearization of the Cauchy stress with respect to the gradient of u. The elasticpart of the Cauchy stress TTTe is assumed to be a linear function of the linearizedstrain tensor DDD(u) := 1

2 (∇u + (∇u)T ) and the viscous part of the Cauchy stressto be a linear function of the symmetric part of the velocity gradient DDD(ut) :=12 (∇ut+(∇ut)T ). In this setting the system (1.1) forms a linear system of equationsand by standard results for linear systems one can establish the existence of a uniquesmooth solution (provided that data are smooth). On the other hand, recently,Rajagopal [17] (see also [18]) has considered generalizations of the classical Kelvin-Voigt model wherein he allows for both the elastic solid and viscous fluid to bedescribed through implicit constitutive relations. Then after a linearization (butnow in a different setting) the author developed a class of models where, the elasticand viscous part of the Cauchy stress are given by general formula

TTTe = HHH(DDD(u)) ,(1.5)

TTTv = GGG(DDD(ut)) ,(1.6)

where HHH,GGG : R2×2sym → R2×2

sym are continuous mappings. This is also the model we

are interested in here. In addition, we assume that HHH,GGG ∈ C0,1(R2×2sym), GGG(0) = 0,

HHH(0) = 0, and that there exist r ∈ [2,∞) and positive constants ν0, ν1 and ν2 suchthat

ν0(1 + |DDD|2)r−22 |BBB|2 ≤ ∂GGG(DDD)

∂DDD: BBB⊗BBB ≤ ν1(1 + |DDD|2)

r−22 |BBB|2,(1.7) ∣∣∣∣∂HHH(DDD)

∂DDD

∣∣∣∣ ≤ ν2(1.8)

for all BBB ∈ R2×2sym and almost all DDD ∈ R2×2

sym. The prototypical example of the model,we are interested in, is given by

(1.9) GGG(DDD) = (1 + |DDD|2)r−22 DDD, HHH(DDD) = (1 + |DDD|2)

q−22 DDD

with some r ≥ 2, q ∈ (1, 2]. For (1.9) it is easy to verify (1.7) and (1.8). Notethat (1.8) allows one to consider more general examples than that introduced in

REGULARITY TO GENERALIZED KELVIN-VOIGHT MODEL 3

(1.9). It is worth noticing that (1.8) says only that HHH is uniformly Lipschitz con-tinuous but does not require any additional structure assumption as potentiality ormonotonicity.

Next, we introduce the assumption put on the data of (1.1). For the density %0,we assume that there are 0 < %∗ ≤ %∗ <∞ such that

(1.10) %0 ∈ L∞, %∗ ≤ %0(x) ≤ %∗ for a.a. x ∈ Ω.

Concerning the density of the external body forces we prescribe

(1.11) %0f ∈(Lr(0, T ; (W 1,r

per)2)∗.

Finally, for the initial displacement and the initial velocity we assume that in ad-dition to (1.3) they also satisfy

u0 ∈ (W 1,2per)

2, v0 ∈ (L2)2.(1.12)

The existence of a weak solution for the problem (1.1) with nonlinear TTTv satis-fying (1.7) with r = 2 can be found in [10], [4] and [20] under certain structuralassumptions on TTT that are more general than (1.4), (1.5) and (1.6). Next, theexistence theory was extended for r ≥ 2 in [3], where the authors assumed thatthe Cauchy stress satisfies (1.4)–(1.8). In addition they showed the uniqueness ofa solution

(1.13) u ∈W 1,∞(0, T ; (L2)2) ∩W 1,r(0, T ; (W 1,r)2)

to (1.1). Although all results in [3] treat the case of mixed boundary conditions,the method presented there works also in the easier periodic case in which we areinterested in here. Moreover, assuming that the data are smooth, one can prove bythe method introduced in [3] that the unique solution u to (1.1) satisfies

(1.14)

(1 + |DDD(ut)|)r−22 DDD(∇ut) ∈ L2(0, T ; (L2)2×2×2),

(1 + |DDD(ut)|)r−22 DDD(utt) ∈ L2(0, T ; (L2)2×2),

utt ∈ Lr′(0, T ; (Lr

′)2).

The main result of the paper is that we improve (1.14) and get the Holdercontinuity of the velocity gradient. Consequently, we use such information to obtainthat the unique weak solution is in fact a classical one provided that the data aresufficiently smooth. The first improvement is established in the following

Theorem 1.1. Let r ≥ 2, T > 0 be arbitrary. Assume that TTT satisfies (1.4)–(1.8)and the data (%0,f ,u0,v0) satisfy (1.3), (1.10)–(1.12). In addition, assume thatthere is p > 2 such that the data fulfill

(%0,u0,v0) ∈W 1,pper × (W 2,p

per)2 × (W 2,p

per)2,

f ∈W 1,2(0, T ; (Lp)2).(1.15)

Then there exists some s > 2 such that

(1.16) ut ∈W 1,∞(0, T ; (Ls)2) ∩ L∞(0, T ; (W 2,sper)

2).

Consequently, there exists α ∈ (0, 1) such that

(1.17) ∇ut ∈ (C0,α(Q))2.

As a consequence of Theorem 1.1 we obtain


Theorem 1.2. Let r ≥ 2 and all assumptions of Theorem 1.1 hold. In additionassume that for some p > 2

(%0,u0,v0) ∈W 1,∞ × (W 3,pper)

2 × (W 3,pper)

2,

f ∈ Lp(0, T ; (W 1,pper)

2), GGG,HHH ∈ C1,1loc (R2×2

sym)2×2.(1.18)

Then the unique strong solution from Theorem 1.1 satisfies

(1.19) ∇ut ∈W 1,p(0, T ; (Lp)2) ∩ Lp(0, T ; (W 2,pper)

2).

As an immediate consequence of Theorem 1.2 and an interpolation Lemma A.2we get

Corollary 1.1. Let r ≥ 2 and all assumptions of Theorem 1.2 hold with somep > 4 and f ∈ C(Q). Then the unique weak solution u is a classical one.

For general systems of partial differential equations Holder continuity of weaksolutions is a rare phenomenon, that can be obtained only under special circum-stances. One of them is that if Ω ⊂ R2 is as in Theorem 1.1. As far as we knowthe only former result in this direction for the problem (1.1) is the one from [9]where Theorem 1.1 is proved in the case r = 2. Another special condition whenregularity (1.17) can be obtained is a special structure of the elliptic term TTTv, see[5], [6]. In these results the structure of the elliptic term is very similar to the formsuggested in (1.9) but the symmetric gradient is replaced with the full gradient,i.e. TTTv = GGG(∇ut). It is a remarkable fact that the method from [5], [6] cannot beapplied in the situation of (1.6), i.e. if the elliptic term depends only on DDD(ut).According to our best knowledge no results about the Holder continuity of gradientsof weak solutions are known if Ω ⊂ Rd, d > 2 and the elliptic part of the equationdepends only on the symmetric part of ∇ut.

The method of the proof of Theorem 1.1 is based on the fact that a smallimprovement of regularity in (1.14) gives Holder continuity of ∇ut. This was firstobserved in [2] and [13] in the stationary case and extended to parabolic systemsin [15] and [8]. This method was used in [9] to prove Theorem 1.1 for r = 2. Firstthe integrability of utt was improved and then the system was treated as an ellipticone on time levels. This method must be modified if r > 2 as in this case we do notknow how to get separately only information about utt. Regularity of utt and ∇uttmust be dealt with simultaneously as it was suggested for generalized Navier-Stokessystem in [11]. This is also the approach that we adopt here to prove Theorem 1.1.

The paper has the following structure. In the next section we introduce someauxiliary lemmas about linear stationary and parabolic systems with bounded mea-surable coefficients. The proofs can be found in the Appendix. In Section 3 weprovide the proof of Theorem 1.1 if r = 2. This result is not new, but it is a basisfor the analysis in Section 4 where Theorem 1.1 is proved for r > 2. Finally, wepresent a sketch of the proof of Theorem 1.2 in Section 5.

In the paper we use standard notation for Lebesgue and Sobolev spaces andtheir norms. If the domain on which the functions are considered is Ω = (0, 1)2, weshorten the notation and write only W 1,q and Lq. The subscript per denotes peri-

odicity with respect to Ω. Particularly, W 1,qper are spaces of functions from W 1,q

loc (R2)for which there is a representative that is periodic with respect to Ω. Moreover,scalar-, vector- and tensor-valued functions are denoted by small letters, small boldletters and bold capital letters in what follows. Also in order to distinguish betweenscalars, vectors and tensors we use the abbreviations Xd and Xd×d for vector- and


tensor-valued function in a Banach space X. For a function GGG : R2×2sym → R2×2 we

denote its gradient ∂DDDGGG. Then for any BBB,DDD ∈ R2×2sym we denote by ∂DDDGGG(DDD) : BBB ⊗BBB

a scalar product of the matrices ∂DDDGGG(DDD) and BBB⊗BBB.

2. Auxiliary results

In this section we recall some results for a linear system similar to (1.1), theproof of these results can be found in the Appendix. This linear system will playa crucial role in the proof of Theorem 1.1, where it will be used as the comparisonproblem. The first auxiliary result is the following lemma.

Lemma 2.1. Let T > 0 be given and assume that AAA : (0, T ) × Ω → R2×2×2×2 isa measurable tensor-valued function satisfying for some 0 < λ1 ≤ λ2 < ∞ and foralmost all (t, x) ∈ (0, T )× Ω the following symmetry and ellipticity conditions

AAAklij (t, x) = AAAijkl(t, x) = AAAjikl(t, x) for all i, j, k, l = 1, . . . , 2,(2.1)

λ1|DDD|2 ≤2∑

i,j,k,l=1

AAAijkl(t, x)DDDijDDDkl ≤ λ2|DDD|2 for all DDD ∈ R2×2sym.(2.2)

Then for any FFF ∈ L2(0, T ; (L2)2×2) and any Ω-periodic w0 ∈ (L2)2 having zeromean value, a unique Ω-periodic weak solution

w ∈ C(0, T ; (L2)2) ∩ L2(0, T ; (W 1,2)2),

∫Ω

w(t, x) dx = 0

exists to the problem

(2.3)wt − div (AAADDD(w)) = −divFFF in (0, T )× Ω,

w(0, ·) = w0(·) in Ω.

Moreover, there exist positive constants K,L > 0 that are independent of T , AAA andFFF such that for all s satisfying

(2.4) 2 ≤ s ≤ 2 +Lλ1

λ2

the following estimate holds

(2.5) supt∈(0,T )

‖w(t)‖2s ≤ K(

1

λ1‖FFF‖2L2(0,T ;(Ls)2×2) + ‖w0‖2s

).

In case one replaces DDD(w) by ∇w in Lemma 2.1, the statement was proved in[15]. However, following the same procedure as in [15] almost step by step one canprove Lemma 2.1 in full generality, see the appendix for detailed proof.

Note that in the previous lemma we did not improve the estimate for the gradientof the solution. As it is usual in parabolic equations the information on the spatialgradient of the solution will be deduced by comparing the equation with its steadyform. Therefore, we recall the following lemma, see for example [14].

Lemma 2.2. Let AAA : Ω→ R2×2×2×2 be measurable tensor-valued function satisfy-ing for some 0 < λ1 ≤ λ2 <∞ and for almost all x ∈ Ω

AAAklij (x) = AAAijkl(x) = AAAjikl(x) for all i, j, k, l = 1, . . . , 2,(2.6)

λ1|DDD|2 ≤2∑

i,j,k,l=1

AAAijkl(x)DDDijDDDkl ≤ λ2|DDD|2 for all DDD ∈ R2×2sym.(2.7)


Then for any FFF ∈ (L2)2×2, there exists a unique Ω-periodic weak solution w ∈(W 1,2)2 such that

∫Ωw dx = 0 solving the problem

(2.8) −div (AAADDD(w)) = −divFFF in Ω.

Moreover, there exist K,L > 0 independent of T , AAA and FFF such that for all

2 ≤ s ≤ 2 +Lλ1

λ2

it holds

(2.9) ‖DDD(w)‖s ≤K

λ1‖FFF‖s.

In general the constants K, L from Lemma 2.1 and Lemma 2.2 may be differentbut without loss of generality we assume in what follows that they are the same.

3. Proof of Theorem 1.1 - case r = 2

This section is devoted to the proof of Theorem 1.1 for r = 2. First, we introducean ε - approximation to the problem (1.1), but we still write u instead of uε for itssolution.

(3.1) ρ0utt − div(GGG(DDD(ut))) = ρ0f + div(HHH(DDD(u ? ωε)) in (0, T )× Ω,

with periodic boundary condition and initial data (u0,v0). Here ω : R2 → R isa standard regularizing kernel, i.e., ω ∈ C∞0 (U(0, 1)) is nonnegative, radially sym-metric,

∫R2 ω dx = 1, and we define ωε(x) = ε−2ω(x/ε). Note that the convolution

in the last term of (3.1) is taken only in space direction. Next, we formulate theexistence result for (3.1), that is the starting point of our analysis.

Lemma 3.1. Let HHH and GGG satisfy (1.7)–(1.8). Assume that %0, f , u0 and v0

satisfy (1.3) and (1.10)–(1.12). In addition, assume that f ∈ W 1,2(0, T ; (L2)2),ρ0 ∈ W 1,2+δ

per for a certain δ > 0 and u0,v0 ∈ (W 2,2per)

2. Then for any ε > 0 thereexists a unique Ω-periodic weak solution u to (3.1), (1.1)2–(1.1)3 that obeys thefollowing a priori estimate∥∥∇2u

∥∥L∞(0,T ;L2)

+ ‖∇ut‖L∞(I,L2) + ‖∇ut‖L2(I,W 1,2) + ‖utt‖L∞(I,L2)

+ ‖utt‖L2(I,W 1,2) ≤ C1(1 + ‖GGG(DDD(v0))‖1,2 + ‖HHH(DDD(u0))‖1,2),(3.2)

where C1 > 0 is independent of ν1 and ε. Moreover, this solution converges to theunique solution of (1.1) as ε→ 0+.

Proof. The proof is presented in [3] for the system (1.1) with mixed (Dirichlet andNeumann) boundary conditions. In our situation smoothing of the term with HHH in(3.1) simplifies the situation and also the periodic boundary conditions are simplerto deal with. Since the proof of Lemma 3.1 follows [3, Theorem 4.1, page 9] line byline we think that it is not necessary to present it here.

Lemma 3.2. Let all the assumptions of Lemma 3.1 hold, ε > 0 be arbitrary andu be the unique weak solution to (3.1). Moreover, assume that for some δ > 0 ands satisfying

(3.3) 2 ≤ s ≤ 2 + min

(Lν0%∗ν1%∗

,δ

2

),


the data fulfill

(u0,v0, %0) ∈ (W 2,sper)

2 × (W 2,sper)

2 ×W 1,2+δper (Ω),

f ∈W 1,2(0, T ; (Ls)2).(3.4)

Then the following estimate holds

supt∈(0,T )

‖utt‖s ≤ (1 + ν1)C(u0,v0, δ,f , ν0, ν2).(3.5)

Proof. First, we construct an Ω-periodic FFF having zero mean value over Ω such that

divFFF = %0f in (0, T )× Ω.

Note that such a construction is possible due to the compatibility condition (1.3).Moreover, using the theory for the divergence equation (see for example [7, 16])and (1.10) we have that

(3.6) ‖FFF‖W 1,2(0,T ;W 1,s(Ω)2×2) ≤ C‖%0f‖W 1,2(0,T ;(Ls)2) ≤ C,

where the last inequality follows from (3.4). Next, we denote w := %0utt andapplying ∂t to (3.1) (note that having (3.2), such procedure is rigorous) we see thatw is a weak solution of the system

(3.7) wt − div (AAADDD(w)) = div FFF in (0, T )× Ω,

where,

AAA :=1

%0

∂GGG(DDD(ut))

∂DDD,

FFF := FFFt +∂HHH(DDD(u ? ωε))

∂DDDDDD(ut ? ω

ε)−[∂GGG(DDD(ut))

∂DDD

](∇%0

%0⊗ utt

).

Since GGG is assumed to satisfy (1.7) with r = 2, and %0 satisfies (1.10), we see thatthe matrix AAA fulfills (2.1)–(2.2) with

(3.8) λ1 :=ν0

%∗and λ2 :=

ν1

%∗.

Hence, assuming that s satisfies (3.3), it also satisfies s ∈ [2, 2 + Lλ1/λ2] and wecan use Lemma 2.1 to deduce that

(3.9) supt∈(0,T )

‖w(t)‖2s ≤K

λ1

∫ T

0

‖FFF‖2s dt+ ‖w(0)‖2s.

We check that the right hand side is finite. To see this, we first evaluate the initialvalue w(0). Using (3.1) we see that

w(0) := %0utt(0) = div (GGG(DDD(v0)) +HHH(DDD(u0 ? ωε))) + %0f(0)

and by using (1.7)–(1.8) and (3.4) we obtain that (for estimating f we use the

embedding W 1,2(0, T ) → C0, 12 ([0, T ]) → C([0, T ]) in 1 - d.)

‖w(0)‖s ≤ ‖divGGG(DDD(v0))‖s + ‖divHHH(DDD(u0 ? ωε))‖s + ‖%0f(0)‖s

≤ ν1‖v0‖2,s + ν2‖u0‖2,s + %∗‖f(0)‖s ≤ C(1 + ν1).(3.10)


It remains to estimate the norm of FFF appearing on the right hand side of (3.9).Using (3.6) and (1.7)–(1.8) we obtain that∫ T

0

‖FFF‖2s dt ≤∫ T

0

‖FFFt‖2s + ν2‖DDD(ut)‖2s +∥∥%−1

0 ∂DDDGGG(DDD(ut))∇%0 ⊗ utt∥∥2

sdt

≤∫ T

0

‖FFFt‖21,s + ν2‖DDD(ut)‖2s +ν2

1

%2∗‖∇%0‖22+δ‖utt‖2 2+δ

2+δ−sdt

≤ C(v0,u0,f , %0, ν2) + C(%0, δ)ν21

∫ T

0

‖utt‖21,2 dt.

(3.11)

Consequently, using the uniform estimate (3.2) we can bound the last term on theright hand side of (3.11) and inserting this and (3.10) into (3.9) we deduce (3.5).

Since, we already know that utt belongs to a better space than L2 uniformly intime, we can improve the spatial regularity of u with help of Lemma 2.2.

Lemma 3.3. Let all assumptions of Lemma 3.1 hold. Then for any ε > 0, δ > 0and s > 0 fulfilling (3.3) and any data satisfying (3.4), the unique solution u to theproblem (3.1) satisfies for a.a. t ∈ (0, T ) the following estimate1

(3.12) ‖∇2ut(t)‖s ≤ C(1 + ‖utt‖L∞(0,T ;Ls) + ‖∇2(u(t) ? ωε)‖s),

with C depending only on (%0,f ,v0,u0, ν0, ν2, T ).

Proof. Since we know from Lemma 3.1 that the equation (3.1) holds pointwisely atalmost all time levels t ∈ (0, T ). We fix such an arbitrary t ∈ (0, T ) and rewrite theproblem (3.1) as

(3.13) −div(GGG(DDD(ut))) = div (FFF +HHH(DDD(u ? ωε))− FFF0) in Ω,

where FFF0 is found such that (note that t is fixed in what follows)

(3.14) divFFF0 = %0utt in Ω, ‖FFF0(t)‖1,s ≤ C%∗‖utt(t)‖sand FFF satisfies

(3.15) divFFF = %0f in Ω, ‖FFF0(t)‖1,s ≤ C%∗‖f(t)‖s.

Next, we fix k ∈ 1, 2, denote w := ∂kut and

AAA :=∂GGG(DDD(ut))

∂DDD.

and differentiate (3.13) in the weak sense with respect to xk. We obtain the follow-ing system of equations

(3.16) − div (AAADDD(w)) = div (∂kFFF + ∂kHHH(DDD(u ? ωε))− ∂kFFF0) in Ω

equipped with periodic boundary conditions and requiring zero mean value for w.Similarly as in the proof of Lemma 3.2 AAA satisfies the assumption of Lemma 2.2with λ1 := ν0 and λ2 := ν1. Hence for any s∗ ∈ [2, 2+Lν0/ν1] we have the estimate

(3.17) ‖DDD(w)‖s∗ ≤K

ν0(‖FFF‖1,s∗ + ‖FFF0‖1,s∗ + ‖HHH(DDD(u ? ωε))‖1,s∗) .

1Note that the right hand side of (3.12) is finite, since for the time derivative we have anestimate due to Lemma 3.2 and the last term in (3.12) is finite due to regularization.


Since, we know that s ≤ 2 + L(ν0%∗)/(ν1%∗) ≤ 2 + Lν0/ν1, we see that (3.17) also

holds for s∗ := s. Moreover, since it holds for any k = 1, 2 we can deduce from(3.17) by using the definition of FFF and FFF0 that

(3.18) ‖∇2ut(t)‖s ≤C(%0)

ν0(‖f(t)‖s + ‖utt(t)‖s + ν2‖DDD(u ? ωε))‖1,s) .

Consequently, using (3.4), (3.3) and the a priori uniform estimates (3.2), we deduce(3.12).

Having all previous estimates, we are ready to prove Theorem 1.1 for r = 2.Since, the case r = 2 will be used in the proof of Theorem 1.1 for r > 2 weformulate it as a special Theorem where we trace the important constant ν1.

Theorem 3.1. Let T > 0 be arbitrary. Assume that TTT satisfies (1.4)–(1.8) withr = 2 and that data (%0,f ,u0,v0) satisfy (1.3), (1.10)–(1.12). In addition, assumethat there is p > 2 such that the data fulfill

(u0,v0, %0) ∈ (W 2,pper)

2 × (W 2,pper)

2 ×W 1,pper(Ω),

f ∈W 1,2(0, T ; (Lp)2).(3.19)

Then there exists a constant C depending only on (%0,v0,u0,f , T, ν0, ν2, p) suchthat for any s fulfilling

(3.20) 2 ≤ s ≤ 2 + min

(Lν0%∗ν1%∗

,p− 2

2

)the unique weak solution (1.1) satisfies the following estimate

(3.21)∥∥∇2ut

∥∥L∞(0,T ;Ls)

≤ C(1 + ν1).

Proof. To prove the theorem it is enough to show estimate (3.21) for the unique solu-tions of the approximating problem (3.1). Indeed, having uniform (ε-independent)estimate (3.21) for the solution of the approximate problem it is easy to let ε→ 0+and to obtain a solution of the original problem (1.1). The estimate (3.21) is validfor this solution due to the weak∗-lower semicontinuity of the norm in L∞(0, T ;Ls).Uniqueness of the solution follows by the method of [3], compare Lemma 3.1. Dueto our assumption on the data and s we see that also all assumptions of Lem-mas 3.1–3.3 are satisfied. We can use (3.12) to prove (3.21). To do so, we need toestimate the last two terms on the right hand side of (3.12). Note that both of themare finite, so we directly have an estimate of the form (3.21) but with right handside depending on ε. To avoid this dependence we estimate both terms as follows.We start with the time derivative for which we obtain by direct use of Lemma 3.2that

(3.22) ‖utt‖L∞(0,T ;Ls) ≤ C(1 + ν1).

Next, for the second term, we get by (3.19)

‖∇2(u(t) ? ωε)‖s ≤ C‖∇2u(t)‖s = C

∥∥∥∥∫ t

0

∇2ut(τ) dτ +∇2u0

∥∥∥∥s

≤ C(

1 +

∫ t

0

‖∇2ut(τ)‖s dτ).

(3.23)


Using (3.22) and (3.23), we see that (3.12) reduces to

‖∇2ut(t)‖s ≤ C(

1 + ν1 +

∫ t

0

‖∇2ut(τ)‖s dτ).

Consequently, applying Gronwall’s lemma in its integral form, we deduce (3.21).

4. Proof of the main theorem-case r > 2

This section is devoted to the proof of Theorem 1.1 for r > 2. It is based on adirect application of the result from the previous section onto a suitable approxi-mating problem.

First, we introduce an quadratic approximation of the problem (1.1). For anyλ > 1 we define Lipschitz continuous functions ηλ and µλ as follows

ηλ(s) :=

1 for s ∈ [0, 2λ2],

− s− 3λ2

λ2for s ∈ (2λ2, 3λ2),

0 for s ≥ 3λ2,

(4.1)

µλ(s) :=

0 for s ∈ [0, λ2],

γλs− λ2

λ2for s ∈ (λ2, 2λ2),

γλ for s ≥ 2λ2,

(4.2)

with some constant γλ ∈ R+ that will be specified later. We approximate GGG by GGGλ

as

(4.3) GGGλ(DDD) := ηλ(|DDD|2)GGG(DDD) + µλ(|DDD|2)DDD.

The most important properties of this approximation are introduced in the followinglemma.

Lemma 4.1. Let GGG satisfy the assumption (1.7) with r > 2 and ν0, ν1 > 0. Letλ > 1 be arbitrary. We set in (4.1) and (4.2)

(4.4) γλ := 7ν1(1 + 3λ2)r−22 .

Then for all BBB ∈ R2×2sym and almost all DDD ∈ R2×2

sym it holds

(4.5) ν0|BBB|2 ≤ ∂DDDGGGλ(DDD) : BBB⊗BBB ≤ ν1|BBB|2

with ν0 and ν1 given as

ν0 := ν0,(4.6)

ν1 := ν1(λ) := 36ν1(1 + 3λ2)r−22 .(4.7)

Moreover, denoting λ(DDD) := min(λ, |DDD|), we get

(4.8) ν0(1 + λ(DDD)2)r−22 |BBB|2 ≤ ∂DDDGGGλ(DDD) : BBB⊗BBB ≤ 36ν1(1 + 3λ(DDD)2)

r−22 |BBB|2

Proof. In order to shorten the notation we denote ∂DDDGGGλ(DDD) : BBB⊗BBB only as ∂DDDGGG(DDD) :

BBB⊗BBB.Using the definition of GGGλ we get

I = ηλ(|DDD|2)∂DDDGGG(DDD) : BBB⊗BBB + 2η′λ(|DDD|2)(DDD ·BBB)(GGG(DDD) ·BBB)

+ µλ(|DDD|2)|BBB|2 + 2µ′λ(|DDD|2)(DDD ·BBB)2.


¿From this identity, the definition of ηλ and µλ we finally conclude that

I =

∂DDDGGG(DDD) : BBB⊗BBB if |DDD|2 < λ2,

∂DDDGGG(DDD) : BBB⊗BBB + γλ|DDD|2 − λ2

λ2|BBB|2 +

2γλλ2

(DDD ·BBB)2 if |DDD|2 ∈ (λ2, 2λ2),

− |DDD|2 − 3λ2

λ2∂DDDGGG(DDD) : BBB⊗BBB

− 2

λ2(DDD ·BBB)(GGG(DDD) ·BBB) + γλ|BBB|2

if |DDD|2 ∈ (2λ2, 3λ2),

= γλ|BBB|2 if |DDD|2 > 3λ2.

Now we remark that by the assumption (1.7) on GGG we get

(GGG(DDD) ·BBB) ≤ ν1(1 + |DDD|2)r−22 |BBB||DDD|.

Defining Y := I/|BBB|2, it follows (note that λ > 1)

ν0(1 + |DDD|2)r−22 ≤ Y ≤ ν1(1 + |DDD|2)

r−22 if |DDD|2 < λ2,

ν0(1 + |DDD|2)r−22 ≤ Y ≤ ν1(1 + |DDD|2)

r−22 + 5γλ if |DDD|2 ∈ (λ2, 2λ2),

γλ − 6ν1(1 + |DDD|2)r−22 ≤ Y ≤ γλ + 6ν1(1 + |DDD|2)

r−22 if |DDD|2 ∈ (2λ2, 3λ2),

γλ = Y if |DDD|2 > 3λ2

and we see that (4.5)–(4.8) follows.

Next, we find λ0 > 0 such that min(Lν0%∗ν1%∗

, p−22

)= Lν0%∗

ν1%∗and such that ν1 ≥ 1

for all λ > λ0.Finally, for arbitrary fixed λ > λ0, we consider an approximation of (1.1) being

of the form

(4.9)

%0utt − divGGGλ(DDD(ut)− divHHH(DDD(u)) = %0f in Q,

u(0, ·) = u0(·) in Ω,

ut(0, ·) = v0(·) in Ω,∫Ω

%0(x)u(t, x) dx =

∫Ω

%0(x)ut(t, x) dx = 0 for all t ∈ (0, T )

equipped with periodic boundary conditions for u.According to Lemma 4.1, GGGλ satisfies all assumptions of Theorem 3.1 and we get

that for all s satisfying2

(4.10) 2 ≤ s ≤ 2 + min

(Lν0%∗ν1%∗

,p− 2

2

)= 2 +

Lν0%∗ν1%∗

the unique weak solution (4.9) satisfies the following estimate∥∥∇2ut∥∥L∞(0,T ;Ls)

≤ C(%0,f , T, ν0, ν2,v0,u0, p)(ν1 + 1)

≤ C(%0,f , T, ν0, ν2,v0,u0, p)ν1.(4.11)

Using the definition of ν0 and ν1 we can set

(4.12) 2 ≤ s = 2 +Rλ2−r

2The constant p > 2 appears in Theorem 1.1 and it is assumed to be the same as in Theorem 3.1.


for a fixed

R ∈ (0, Lρ∗ρ∗

ν0

10ν1(1

3)

2−r2 )

and rewrite the estimate (4.11) as

(4.13)∥∥∇2ut

∥∥L∞(0,T ;Ls)

≤ C(%0,f , T, ν0, ν2,v0,u0, p, r)λr−2.

Our main goal, based on estimate (4.13), is to find a sufficiently large λ > λ0 suchthat

(4.14) M := ‖1 + λ(DDD(ut))2‖L∞(0,T ;L∞) ≤ λ2.

For such λ the equality GGGλ(DDD(ut)) = GGG(DDD(ut)) holds a.e. in (0, T )× Ω and conse-quently, u solves the original problem (1.1).

We start with estimates uniform with respect to λ. In the following the positiveconstant C is always independent of λ but it can depend on the data(f , %0,u0,v0, p, r, ν0, ν1, ν2). From Lemma 3.1 we know that∥∥∇2u

∥∥L∞(0,T ;L2)

+ ‖utt‖L∞(0,T ;L2) ≤ C(1 +∥∥GGGλ(DDD(v0))

∥∥1,2

).(4.15)

This estimate is still λ-dependent. However, using the definition of GGGλ and theassumptions on the data (1.15), we see that∥∥GGGλ(DDD(v0))

∥∥1,2≤ C(1 +

∥∥|DDD(v0)|r−1∥∥

2+∥∥|DDD(v0)|r−2|∇2v0|

∥∥2) ≤ C(1 + ‖v0‖r−1

2,p ),

where for the last inequality we used the Holder inequality and the embeddingW 2,p →W 1,∞ (valid for p > 2). Consequently, we see that (4.15) can be rewrittenas

(4.16)∥∥∇2u

∥∥L∞(0,T ;L2)

+ ‖utt‖L∞(0,T ;L2) ≤ C.

Uniform estimates on ∇2ut are obtained by the same method as in the proof ofLemma 3.3. We rewrite (4.9) for a.e. t ∈ (0, T ) as

− divGGGλ(DDD(ut(t))) = %0f(t) + divHHH(DDD(u(t)))− %0utt(t).

This equation holds pointwisely in Ω due to (4.11) and it is allowed to test it withut(t) and −∆ut(t). Doing so, one gets with help of (4.16) and (4.8) that∫

Ω

(1 + λ(DDD(ut(t)))2)

r−22 |∇2ut(t)|2 ≤ C

and by a simple algebraic manipulation we deduce that

(4.17)∥∥(1 + λ(DDD(ut))

2)r4

∥∥L∞(0,T ;W 1,2)

≤ C.

Finally, we combine the non-uniform estimate (4.13) with the uniform ones (4.16)and (4.17) to deduce (4.14). First, for any s ∈ (2, s) and α ∈ (0, 2) such that1 = α/2 + (s− α)/s, i.e.

(4.18) s− 2 = (2− α)(s− 2)/2

we use the Holder inequality to get∥∥(1 + λ(DDD(ut))2)

r4

∥∥sL∞(0,T ;W 1,s)

≤ C∥∥(1 + λ(DDD(ut))

2)r4

∥∥αL∞(0,T ;W 1,2)

∥∥(1 + λ(DDD(ut))2)

r4

∥∥s−αL∞(0,T ;W 1,s)

.


To estimate the term on the right-hand side, we use the definition of λ, the uniformestimate (4.17) and the non-uniform estimate (4.13) to conclude that

(4.19)

∥∥(1 + λ(DDD(ut))2)

r4

∥∥sL∞(0,T ;W 1,s)

≤ C(1 +∥∥∇(1 + λ(DDD(ut))

2)r4

∥∥s−αL∞(0,T ;Ls)

)

≤ C(1 +∥∥∇2ut

∥∥s−αL∞(0,T ;Ls)

λ(r−2)(s−α)

2 )

≤ Cλ(r−2)(s−α) 32 .

Finally, we focus on finding such λ > λ0 so that (4.14) holds. Using the embed-ding theorem W 1,s → L∞ with the precise embedding constant (see [22, Proof ofTheorem 2.4.1]), the definition (4.14) of M and the estimate (4.19), we get that

Mr4 ≤

(C

s− 2

)1− 1s ∥∥(1 + λ(DDD(ut))

2)r4

∥∥L∞(0,T ;W 1,s)

≤(

C

s− 2

)1− 1s

λ3(r−2)(s−α)

2s .

(4.20)

Hence, to show (4.14) and consequently to finish the proof, it is enough to findλ > λ0, s ∈ (2, s) and α ∈ (0, 2) fulfilling (4.18) such that

(4.21)

(C

s− 2

)1− 1s

λ3(r−2)(s−α)

2s ≤ λ r2 .

Next, using (4.12) and (4.18) it is not difficult to observe the following identities(C

s− 2

)1− 1s

λ(r−2) 32s−αs =

(2C

(2− α)(s− 2)

)1− 1s

λ(r−2) 32s−αs

=

(2C

(2− α)R

)1− 1s

λ(r−2)(1− 1s+ 3

2s−αs )

and we see that (4.21) is equivalent to

(4.22)

(2C

(2− α)R

)1− 1s

λ(r−2)(1− 1s+ s−α

s32 ) ≤ λ r2 .

Since, limα→2− s = 2 we have

limα→2−

(r − 2)(1− 1

s+s− αs

3

2

)=r − 2

2<r

2

and therefore it is always possible to find α ∈ (0, 2) (and consequently s) and ε > 0such that

r

2− (r − 2)

(1− 1

s+s− αs

3

2

)> ε.

Thus, we fix such α and s and we see that to fulfill (4.22) it is enough to find λ > λ0

such that (2C

(2− α)R

)1− 1s

≤ λε,

which is clearly possible and therefore the proof of Theorem 1.1 for the case r > 2is complete.


5. Proof of Theorem 1.2

In this section we provide only formal a priori estimates. However, they can bemade rigorous by the method of Section 3, see the approximation (3.1) and theproof of Theorem 3.1.

Let u be the unique solution constructed in Theorem 1.1. In particular it satisfies(1.16) and (1.17). We define w = ∂kut for fixed k ∈ 1, 2. We differentiate (1.1)with respect to xk and find that w solves the problem

(5.1)∂twj −

2∑i,α,β=1

AAAαβij ∂i∂αwβ = FFFj in Q for j = 1, 2,

w(0, ·) = ∂kv0 in Ω,

with

AAAαβij =1

ρ0∂αβGGGij(DDD(ut)),

FFFj =1

ρ0

( 2∑i=1

[∂2DDDGGGij(DDD(ut)) : (DDD(∂iut)⊗DDD(∂kut)) + ∂DDDHHHij(DDD(u)) : DDD(∂i∂ku)

+ ∂2DDDHHHij(DDD(u)) : (DDD(∂iu)⊗DDD(∂ku))

]− ∂kρ0utt + ∂k(ρ0f)

).

In formulating (5.1) the symmetry of AAA was used. Due to (1.17) and the assumptionon GGG we know that AAA ∈ C0,α(Q) for a certain α ∈ (0, 1) and AAA is elliptic, i.e. thereare λ1, λ2 > 0 such that (2.2) holds. Due to the symmetry of AAA we know that (2.2)holds even for all BBB ∈ R2×2. For FFF and arbitrary σ ∈ (1, p] we obtain from theproperties of GGG, HHH and (1.17)

‖FFF‖σ ≤ C(1 +∥∥∇2ut

∥∥2

2σ+∥∥∇3u

∥∥σ

+ ‖∇u‖22σ + ‖utt‖σ).

The constant C > 0 may depend on HHH, GGG, T , u0, ρ0.Using standard parabolic Lp-theory on Qt = (0, t)×Ω, see [12, Section IV.9] we

obtain since k = 1, 2 is arbitrary

‖∇utt‖σ,Qt +∥∥∇3ut

∥∥σ,Qt≤ C(1 +

∥∥∇2ut∥∥2

2σ,Qt+∥∥∇3u

∥∥σ,Qt

+ ‖∇u‖22σ,Qt + ‖utt‖σ,Qt).

Now we use the inequality∫Ω

|∇3u(t)|σ ≤ C(∥∥∇3u

∥∥σσ,Qt

+∥∥∇3ut

∥∥σσ,Qt

+

∫Ω

|∇3u0|σ)

to get∥∥∇3u(t)∥∥σσ,Ω

+ ‖∇utt‖σσ,Qt +∥∥∇3ut

∥∥σσ,Qt≤

C(1 +∥∥∇2ut

∥∥2σ

2σ,Qt+∥∥∇3u

∥∥σσ,Qt

+ ‖∇u‖2σ2σ,Qt + ‖utt‖σσ,Qt +∥∥∇3u0

∥∥σσ,Ω

).

Due to the assumption on u0 we know that∥∥∇3u0

∥∥σσ,Ω

< +∞ for all σ ≤ p. If∥∥∇2ut∥∥2σ

2σ,Qt+ ‖∇u‖2σ2σ,Qt + ‖utt‖σσ,Qt is bounded we get by Gronwall’s inequality

‖∇utt‖σσ,Qt +∥∥∇3ut

∥∥σσ,Qt

< +∞.


This is always true for σ ∈ (1, s/2], where s > 2 is taken from Theorem 1.1. Bythe following multiplicative inequality [12, Theorem II.2.2] we get that for anymeasurable function z

‖z‖σs+22

σ s+22 ,Q

≤ C ‖z‖σs2

L∞(0,T ;Ls) ‖∇z‖σσ,Q .

If we take into account (5) and (1.16) we find that for σ ∈ (1, s/2]∥∥∇2u∥∥σ s+2

2

+∥∥∇2ut

∥∥σ s+2

2

+ ‖utt‖σ s+22 ,Q < +∞.

Since σ s+22 > 2σ for all σ > 1 we get the statement (1.19) of Theorem 1.2 after

finite number of iterations.

Appendix A. Proof of Lemma 2.1

First, we recall and prove Lemma 2.1. We only sketch the proof of it and focuson main differences compared to the differences to [15], where the full gradient caseis treated.

Lemma A.1. Let T > 0 be given and assume that AAA : (0, T ) × Ω → R2×2×2×2 ismeasurable tensor-valued function satisfying for almost all (t, x) ∈ (0, T ) × Ω thefollowing

AAAklij (t, x) = AAAijkl(t, x) = AAAjikl(t, x) for all i, j, k, l = 1, . . . , 2,(A.1)

λ1|DDD|2 ≤2∑

i,j,k,l=1

AAAijkl(t, x)DDDijDDDkl ≤ λ2|DDD|2 for all DDD ∈ R2×2sym.(A.2)

Then for any FFF ∈ L2(0, T ; (L2)2×2) and any Ω-periodic w0 ∈ (L2)2 having zeromean value, there exists a unique Ω-periodic weak solution

w ∈ C(0, T ;L2) ∩ L2(0, T ;W 1,2)

∫Ω

w(t, x) dx = 0

to the problem

(A.3)wt − div (AAADDD(w)) = divFFF in (0, T )× Ω,

w(0, ·) = w0(·) in Ω.

Moreover, there exists positive constants K,L > 0 that are independent of T , AAA andFFF such that for all s satisfying

(A.4) 2 ≤ s ≤ 2 +Lλ1

λ2

the following estimate holds

(A.5) supt∈(0,T )

‖w‖2s ≤ K

(1

λ1

∫ T

0

‖FFF‖2s dt+ ‖w0‖2s

).

Proof. The existence and uniqueness of a weak solution to (A.3) is standard. There-fore, we focus here only on the proof of (A.5). We provide only the formal proofbut everything can be done rigorously by mollifying AAA and w0, applying standardresults for the heat equation, deriving uniform bounds of the type (A.5) and thenpassing to the limit.

The main idea of the proof is to use w|w|s−2 as a test function in the weakformulation of (A.3). Due to the presence of a nonlinearity in the test functionwe obtain a pollution term coming from the elliptic term. To handle it, we use a


simple inequality that can be deduced by an integration by parts formula. Hence,for any s ≥ 2 and any smooth periodic u we have∫

Ω

|u|s−2|∇u|2 dx = −∫

Ω

|u|s−24u · u dx− (s− 2)

∫Ω

|u|s−2|∇|u||2 dx

= −(s− 2)

∫Ω

|u|s−2|∇|u||2 dx− 2

∫Ω

|u|s−2 div(DDD(u)) · u dx

+

∫Ω

|u|s−2∇(divu) · u dx

= −(s− 2)

∫Ω

|u|s−2|∇|u||2 dx+ 2

∫Ω

|u|s−2|DDD(u)|2 dx

+ 2(s− 2)

∫Ω

|u|s−3DDD(u)u · ∇|u| dx

−∫

Ω

|u|s−2|divu|2 − (s− 2)

∫Ω

|u|s−3 divu∇|u| · u dx

Consequently, moving the term with the good sign to the left hand side we obtainthe inequality∫

Ω

|u|s−2|divu|2 + (s− 2)|u|s−2|∇|u||2 + |u|s−2|∇u|2 dx

≤ 2

∫Ω

|u|s−2|DDD(u)|2 dx+ 2(s− 2)

∫Ω

|u|s−2|DDD(u)||∇|u|| dx

+ (s− 2)

∫Ω

|u|s−2|divu||∇|u|| dx.

Finally, using the pointwise estimate |divu| ≤ C|DDD(u)| and applying the Younginequality we deduce that for any s > 2 there exists Cs > 0 such that

(A.6)

∫Ω

|u|s−2|∇u|2 dx ≤ Cs∫

Ω

|u|s−2|DDD(u)|2 dx.

If we restrict to s ∈ [2, 10] we can find C∗ > 0 such that for all s ∈ [2, 10], Cs < C∗.With estimate (A.6) we can easily continue by using the standard procedure, see

[15], [8]. We test (2.3) by |w|s−2w with arbitrary s ∈ [2, 4] to get

1

s

d

dt‖w‖ss +

∫Ω

|w|s−2AAADDD(w) ·DDD(w) dx = −∫

Ω

FFF · ∇(|w|s−2w) dx

−∫

Ω

AAADDD(w) · (∇|w|s−2 ⊗w) dx.

(A.7)

Consequently, using (A.2) and (A.6) we observe that (we use the Young inequalityto get the second estimate)

1

s

d

dt‖w‖ss + λ1

∫Ω

|w|s−2|DDD(w)|2 dx

≤ C( ∫

Ω

|FFF||w|s−2|∇w| dx+ λ2(s− 2)

∫Ω

|w|s−2|∇w|2 dx)

≤ (Cλ2(s− 2) +λ1

2C∗)

∫Ω

|w|s−2|∇w|2 dx+C2C∗

2λ1

∫Ω

|FFF|2|w|s−2 dx,

≤ λ1

∫Ω

|w|s−2|DDD(w)|2 dx+C2C∗

2λ1

∫Ω

|FFF|2|w|s−2 dx,

(A.8)


provided that

(A.9) CC∗λ2(s− 2) + λ1/2 < λ1.

Thus, defining L := 12CC∗ , we see that for all 2 ≤ s ≤ 2 + Lλ1/λ2 the condition

(A.9) is automatically met and the inequality (A.8) implies that

d

dt‖w‖ss ≤

C

λ1

∫Ω

|FFF|2|w|s−2 dx ≤ C

λ1‖FFF‖2s‖w‖s−2

s

and we finally obtain thatd

dt‖w‖2s ≤

C

λ1‖FFF‖2s

that leads to (A.5) after integration w.r.t. t ∈ (0, T ).

Lemma A.2 (See also [12]). Let T > 0 be arbitrary and p > 4. Then the followingembedding holds

(A.10) W 1,p(0, T ;Lp) ∩ Lp(0, T ;W 2,p) → C([0, T ], C1(Ω)).

Proof. By using an interpolation theorem , see [1, proof of Corollary 4.5 (ii)], wefind that for any α ∈ [0, 1]

W 1,p(0, T ;Lp) ∩ Lp(0, T ;W 2,p) →Wα,p(0, T ;W 2(1−α),p).

Consequently, setting α := 14 and using the standard Sobolev embedding we get

(A.10), provided that p > 4.

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Mathematical Institute, Faculty of Mathematics and Physics, Charles University,Sokolovska 83, 186 75 Praha 8, Czech Republic

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Mathematical Institute, University of Koblenz-Landau, Campus Koblenz, Univer-

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On existence of a classical solution to a generalized ...ncmm.karlin.mff.cuni.cz/preprints/11261134953pr16.pdf · ON EXISTENCE OF A CLASSICAL SOLUTION TO A GENERALIZED KELVIN-VOIGT