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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 407940, 19 pages
http://dx.doi.org/10.5402/2012/407940
Research Article

On Strong Solutions of Regularized Model of a Viscoelastic Medium with Variable Boundary

Department of Mathematics, Voronezh State University, Universitetskaya pl., 1, Voronezh 394 000, Russia

Received 29 September 2011; Accepted 19 October 2011

Academic Editor: Y. Xu

Copyright © 2012 Vladimir Orlov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the initial-value problem for systems of equations describing the evolution of a viscoelastic medium with variable boundary with memory along the trajectories of a velocity field, which generalizes the Navier-Stokes system of equations. Nonlocal existence and uniqueness theorem of strong solutions containing senior square-integrable derivatives in the planar case are established.

1. Introduction

Let Ω𝑡𝑅𝑛, 𝑛=2,3, be a family of bounded domains with 𝜕Ω𝑡𝐶2, depending smoothly on 0𝑡𝑇,<+. Consider the initial-boundary problem 𝜕𝑣+𝑣𝜕𝑡𝑖𝜕𝑣𝜕𝑥𝑖𝜇0Δ𝑣𝜇Div𝑡0exp(𝜆(𝑠𝑡))(𝑣)(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠+𝑝=𝑓(𝑡,𝑥),(𝑡,𝑥)𝑄𝑇,div𝑣(𝑡,𝑥)=0,(𝑡,𝑥)𝑄𝑇,(1.1)𝑣(0,𝑥)=𝑣0(𝑥),𝑥Ω0,𝑣(𝑡,𝑥)=𝜓(𝑡,𝑥),(𝑡,𝑥)𝑆𝑇,(1.2) where 𝑄𝑇={(𝑡,𝑥)0𝑡𝑇,𝑥Ω𝑡}, 𝑆𝑇={(𝑡,𝑥)𝑡[0,𝑇],𝑥𝜕Ω𝑡}.

In (1.1)-(1.2), the velocity 𝑣(𝑡,𝑥)=(𝑣1(𝑡,𝑥),,𝑣𝑛(𝑡,𝑥)) and the pressure 𝑝(𝑡,𝑥) are unknown, 𝑓(𝑡,𝑥) is given outer forces, 𝑣0(𝑥) is prescribed initial velocity, (𝑣) = {𝑖𝑗}𝑛𝑖,𝑗=1 is the rate-of-strain tensor, that is, the matrix with elements 𝑖𝑗(𝑣)=(1/2)(𝜕𝑣𝑖/𝜕𝑥𝑗+𝜕𝑣𝑗/𝜕𝑥𝑖), 𝜇0>0,𝜇0,𝜆0. The divergence Div of a matrix is defined as a vector with elements which are the divergences of the lines. The vector function 𝑧(𝜏;𝑡,𝑥) is a solution to the Cauchy problem (in the integral form) 𝑧(𝜏;𝑡,𝑥)=𝑥+𝜏𝑡[]𝑣(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠,𝜏0,𝑇,(𝑡,𝑥)𝑄𝑇.(1.3) The system describes the evolution Ω𝑡 of a viscoelastic medium Ω0 with a memory along the trajectories of the velocity vector field.

In the case of a cylindrical domain 𝑄𝑇 (Ω𝑡Ω0), the local existence and uniqueness theorem (e.u.t.) for strong solutions (𝑣𝑊𝑞1,2(𝑄𝑇), 𝑞>𝑛) to the problem (1.1)-(1.2), (1.3) and nonlocal e.u.t. for small data were established in [1, 2]. Nonlocal e.u.t. for strong solutions (𝑣𝑊21,2(𝑄)) to the problem (1.1)-(1.2), (1.3), where 𝑣 in (1.3) is replaced by some regularized vector field ̃𝑣 (regularized problem) for 𝑛=2, was established in [3]. There was given the motivation of the regularization of the velocity field 𝑣. In the case that 𝜇=0 and 𝑄𝑇 is noncylindrical domain, the local for 𝑛=3 and nonlocal for 𝑛= 2 (e.u.t.) of strong solutions (𝑣𝑊21,2(𝑄𝑇)) to the system (1.1)-(1.2) were established in [4]. The e.u.t. of weak solutions to the regularized problem (1.1)-(1.2), (1.3) with 𝜇>0 in noncylindrical domain was established in [5].

Our goal is to prove a nonlocal e.u.t. for solutions (𝑣,𝑝), 𝑣𝑊21,2(𝑄𝑇), 𝑝𝑊20,1(𝑄𝑇) by 𝑛= 2. First, we set a priori estimates, then we prove the local solvability by means of reduction of the initial problem to a suitable operator equation, and then we obtain a nonlocal theorem using the obtained a priori estimates.

Next, referring to the solution to the problem (1.1)-(1.2), (2.1), we, as usual, keep in mind only the solenoidal part 𝑣 of the pair (𝑣,𝑝). The function 𝑝 is found easily by the standard way (see, e.g., [6, Chapter  3, page  214]).

The constants in the estimates which do not depend on substantial parameters, we denote by 𝑀, sometimes with indices. We take the summation convention for repeated indices.

2. Notations and Statement of the Results

The norms of a function 𝑢 in real spaces 𝐿2(Ω𝑡), 𝐿2(𝑄𝑇) and in Sobolev spaces 𝑊𝑘2(Ω𝑡), 𝑊2𝑚,𝑘(𝑄𝑇) we denote by |𝑢|0,𝑡, 𝑢0, |𝑢|𝑘,𝑡, 𝑢𝑘,𝑚, respectively. We use the same designation for scalar, vector, or matrix functions.

The index 𝑘 can be negative (see [7, page  251]). We denote by (,)𝑡 the inner product in 𝐿2(Ω𝑡), 𝐻𝑡 and 𝑉𝑡 are the closure of smooth compactly supported solenoidal functions in Ω𝑡 in the norm of 𝐿2(Ω𝑡) and 𝑊12(Ω𝑡), respectively.

We denote by 𝒫𝑡 the orthogonal projector in 𝐿2(Ω𝑡) on 𝐻𝑡. Consider in 𝐻𝑡 unbounded linear operator 𝐴𝑡𝑣=𝒫𝑡Δ𝑣, with the domain 𝐷(𝐴𝑡)=𝑊22(Ω𝑡)𝐻𝑡𝑊12(Ω𝑡). Operator 𝐴𝑡 (see [8, page  54]) is a positive self-adjoint operator. We define the regularizer 𝑆𝛿, 𝛿>0 as 𝑆𝛿𝑆𝑣=𝛿(𝑣𝑤)+𝑤, where 𝑆𝛿𝑣=(𝐼+𝛿𝐴𝑡)1𝑣. Consider the Cauchy problem 𝑧(𝜏;𝑡,𝑥)=𝑥+𝜏𝑡̂[]𝑣(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠,𝜏0,𝑇,(𝑇,𝑥)𝑄𝑇,(2.1) where 𝑆𝛿̂𝑣𝑣. If 𝑣𝑊21,2(𝑄𝑇) and the evolution of Ω𝑡 is smooth, then regularized problem (2.1) is uniquely solvable (see below). As to regularization operator 𝑆𝛿, we note that it is uniformly bounded of 𝑡 as an operator from 𝐿2(Ω𝑡) in 𝐶1(Ω𝑡). There are other constructions of such operators (see, e.g., [3, 9]). Every one of them has the property of strong convergence of 𝑆𝛿 to 𝐼 by 𝛿0.

We believe that Ω𝑡=̃𝑧(𝑡,0,Ω0), where ̃𝑧(𝜏;𝑡,𝑥) is a solution to the Cauchy problem ̃𝑧(𝜏;𝑡,𝑥)=𝑥+𝜏𝑡[]𝑤(𝑠,̃𝑧(𝑠;𝑡,𝑥))𝑑𝑠,𝜏0,𝑇,(𝑡,𝑥)𝑄𝑇,(2.2) where 𝑤 is sufficiently smooth on some neighborhood of 𝑄𝑇 solenoidal function. The boundary condition (1.2) is defined as 𝜓(𝑡,𝑥)=𝑤(𝑡,𝑥), (𝑡,𝑥)𝑆𝑇 under the natural condition 𝑣0(𝑥)=𝜓(0,𝑥), 𝑥𝜕Ω0. Note that the same 𝑄𝑇 admits representation via evolution Ω𝑡 by means of different functions 𝑤 with correspondent boundary values 𝜓.

The main result reads as follows.

Theorem 2.1. Let 𝑛=2, 𝑓𝐿2(𝑄𝑇), 𝑣0𝑊12(Ω0), div𝑣0(𝑥)=0. Then, problem (1.1)-(1.2), (2.1) has a unique solution 𝑣𝑊21,2(𝑄𝑇).

As to the form of 𝑄𝑇 and the boundary function 𝜓(𝑡,𝑥), we preferred to concentrate ourselves on the essence of the problem and chose a simple way to provide the solvability of (2.1) on [0,𝑇] and to avoid a complexity in the proofs. Let the assumption 𝑄𝑇 be a sufficiently arbitrary domain, and let 𝜓(𝑡,𝑥) be the trace of a solenoidal 𝜓(𝑡,𝑥)𝑊21,2(𝑄𝑇) and satisfy Γ(𝜓(𝑡,𝑥),𝑛(𝑥))𝑑𝑥=0 (𝑛(𝑥) is the outward unit normal) (as in [4]) leads to the effect of leakage through the boundary that in turn leads to the necessity to study the properties of solutions to (2.1) which may be defined only on a part of [0,𝑇].

We believe also 𝜆=0,𝜇0=1, and 𝑤 is sufficiently smooth in order to simplify the calculuses (in fact, it is sufficient that 𝑤𝑊𝑞1,2(𝑄),𝑞>𝑛).

We reduce the original problem to a problem with zero boundary data. Let 𝑣=𝑢+𝑤, where a solenoidal function 𝑤𝑊21,2(𝑄) and 𝑤(𝑡,𝑥)=𝑤(𝑡,𝑥) on 𝑆𝑇. Then, 𝑢 satisfies relations 𝜕𝑢+𝑢𝜕𝑡𝑖𝜕𝑢𝜕𝑥𝑖+𝑤𝑖𝜕𝑢𝜕𝑥𝑖+𝑢𝑖𝜕𝑤𝜕𝑥𝑖Δ𝑢𝜇Div𝑡0(𝑢)(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠𝜇Div𝑡0(𝑤)(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠+𝑝=𝐹(𝑡,𝑥),div𝑢(𝑡,𝑥)=0,(𝑡,𝑥)𝑄𝑇,(2.3)𝑢(0,𝑥)=𝑢0(𝑥),𝑥Ω0;𝑢(𝑡,𝑥)=0,(𝑡,𝑥)𝑆𝑇,(2.4) and (2.1). Here, 𝑢=(𝑢1,𝑢2),𝑤=(𝑤1,𝑤2), 𝐹(𝑡,𝑥)=𝑓(𝑡,𝑥)𝜕𝑤𝑤𝜕𝑡𝑖𝜕𝑤𝜕𝑥𝑖+Δ𝑤,𝑢0(𝑥)=𝑣0(𝑥)𝑤(0,𝑥).(2.5) Instead of Theorem 2.1, it will be convenient for us to prove its reformulation for the problem (2.3)-(2.4), (2.1).

Theorem 2.2. Let 𝐹𝐿2(𝑄𝑇), 𝑢0𝑉0. Then, problem (2.3)-(2.4), (2.1) has a unique solution 𝑢𝑊21,2(𝑄𝑇).

The proof of Theorem 2.2 consists of several steps.

3. Properties of Solutions to Cauchy Problem

By a solution to (1.3), we mean a continuous with respect to all variables vector function 𝑧(𝜏;𝑡,𝑥), satisfying (1.3). Denote by 𝑢𝑥 a Jacobi matrix of the vector function 𝑢(𝑥), by 𝑢𝑥𝑥 a tensor which is composed of second derivatives of the components of a vector function 𝑢(𝑥), and by |𝑢𝑥|20 and |𝑢𝑥𝑥|20 the sum of squares of 𝐿2(Ω)-norms of the first and second derivatives, respectively. The following result was established in [1] for a cylindrical domain 𝑄𝑇.

Lemma 3.1 (see [2]). Let 𝑣,𝑣1,𝑣2𝑊𝑞1,2(𝑄𝑇),𝑞>𝑛 and vanish at 𝑆𝑇. Then, the corresponding Cauchy problems (1.3) are uniquely solvable, and the estimates 𝑧1(𝜏;𝑡,𝑥)𝑧2(𝜏;𝑡,𝑥)𝑊1𝑞1(Ω𝑡)||||𝑀𝜏𝑡𝑣1(𝑠,𝑥)𝑣2(𝑠,𝑥)𝑊𝑞1(Ω𝑠)||||𝑀||||𝑑𝑠×exp𝜏𝑡max𝑘=1,2𝑣𝑘(𝑠,𝑥)𝑊2𝑞1(Ω𝑠)||||,[]𝑑𝑠𝜏,𝑡0,𝑇,1𝑞1𝑧𝑞,𝑥𝑥(𝜏;𝑡,𝑥)𝐿𝑞(Ω𝑡)𝑀𝑇0𝑣(𝑠,𝑥)𝑊2𝑞(Ω𝑠)||||𝑑𝑠𝜏𝑡𝑣(𝑠,𝑥)𝑊2𝑞(Ω𝑠)||||𝑑𝑠(3.1) hold. If, moreover, 𝑣(𝑠,𝑥)𝐶1(Ω𝑠)𝐿1[0,𝑇], then 𝑧𝑥(𝜏;𝑡,𝑥)𝐶(Ω𝑡)||||𝑀exp𝜏𝑡𝑣(𝑠,𝑥)𝐶1(Ω𝑠)||||[]𝑑𝑠,𝜏,𝑡0,𝑇.(3.2)

Using the smoothness of the evolution of Ω𝑡 and the fact that the lateral side consists of the trajectories of a smooth velocity field 𝑤, we can show that the similar result holds for noncylindrical domain 𝑄𝑇 by 𝑣,𝑣1,𝑣2=𝜓 on 𝑆𝑇. To do this, one has as in [10, Chapter  3, Section  1.2] to continue 𝑣,𝑣1,𝑣2 on 𝑅𝑛 with the preservation of the class by functions vanishing outside some compact domain 𝑄=[0,𝑇]×Ω,ΩΩ𝑡,0𝑡𝑇, and use the result for the cylindrical case. In virtue of the uniqueness theorem, the solutions to Cauchy problems of our initial problems are the restrictions of solutions of the extended Cauchy problems.

Consider the Cauchy problem (2.1). Best, compared with 𝑣, properties of ̂𝑣 allow to strengthen the result of Lemma 3.1.

Lemma 3.2. Let 𝑣,𝑣1,𝑣2𝑊21,2(𝑄𝑇). Then, corresponding Cauchy problems (2.1) are uniquely solvable, and the estimates 𝑧𝑥(𝜏;𝑡,𝑥)𝐶(Ω𝑡)𝑀||||𝑀exp𝜏𝑡||||𝑣(𝑠,𝑥)1,𝑠||||[]||𝑧𝑑𝑠,𝜏,𝑡0,𝑇,(3.3)1𝑥(𝜏;𝑡,𝑥)𝑧2𝑥||(𝜏;𝑡,𝑥)0,𝑡𝑣𝑀11,2,𝑣21,2||||𝜏𝑡||𝑣1(𝑠,𝑥)𝑣2||(𝑠,𝑥)1,𝑠||||||𝑧𝑑𝑠,(3.4)𝑥𝑥||(𝜏;𝑡,𝑥)0,𝑡𝑀𝑣1,2||||𝜏𝑡||||𝑣(𝑠,𝑥)2,𝑠||||[]𝑧𝑑𝑠,𝜏,𝑡0,𝑇,(3.5)1(𝜏;𝑡,𝑥)𝑧2(𝜏;𝑡,𝑥)𝐶(Ω𝑡)𝑣𝑀11,2,𝑣21,2||||𝜏𝑡||𝑣1(𝑠,𝑥)𝑣2||(𝑠,𝑥)1,𝑠||||,[]𝑑𝑠𝜏,𝑡0,𝑇(3.6) hold.

In the proof of Lemma 3.2, Lemma 3.1 and uniform with respect to 𝑡 estimate ̂𝑣(𝑡,𝑥)𝐶1(Ω𝑡)𝑀|𝑣(𝑡,𝑥)|1,𝑡 are used. The last estimate easily follows from the continuous imbedding 𝑊𝜎2(Ω𝑡)𝐶1(Ω𝑡), 𝜎>2 (see [11, page  408]), and the inclusion 𝐷((𝐼+𝛿𝐴𝑡)1)𝑊22(Ω𝑡).

4. First A Priori Estimate

Theorem 4.1. Let 𝑢 be a solution to the problem (2.3)-(2.4), (2.1). Then, inequality sup𝑡||||𝑢(𝑡,𝑥)0,𝑡+𝑢(𝑡,𝑥)0,1𝑀1𝑤1,2𝐹0,1+||𝑢0||0,0(4.1) holds.

Recall that 𝑣20,1=𝑇0|𝑣(𝑡,𝑥)|20,1𝑑𝑡.

Proof. Let Ψ(𝑡,𝑥) be a sufficiently smooth scalar function. Since Ω𝑡=̃𝑧(𝑡,0,Ω0), where ̃𝑧 is a solution to the Cauchy problem (2.2) for 𝑤 with div𝑤=0, then (see [12, page  8]) 𝑑𝑑𝑡Ω𝑡Ψ(𝑡,𝑥)𝑑𝑥=Ω𝑡𝜕𝜕𝑡Ψ(𝑡,𝑥)𝑑𝑥+𝜕Ω𝑡Ψ(𝑡,𝑥)𝑑𝑥.(4.2) It follows that 𝑑||||𝑑𝑡𝑢(𝑡,𝑥)20,𝑡𝜕=2𝜕𝑡𝑢(𝑡,𝑥),𝑢(𝑡,𝑥)𝑡,(4.3) where Ψ(𝑡,𝑥)=𝑢(𝑡,𝑥)𝑢(𝑡,𝑥), 𝑢𝑊21,2(𝑄𝑇), and vanishes on 𝑆𝑇.
Multiply both sides of (2.3) in 𝐿2(Ω𝑡) by 𝑢. The standard arguments as in [5] and (4.2), (4.3) give12𝑑||||𝑑𝑡𝑢(𝑡,𝑥)20,𝑡+||||𝑢(𝑡,𝑥)21,𝑡=(𝐹(𝑡,𝑥),𝑢(𝑡,𝑥))𝑡+𝜇𝑡0(𝑢)(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠,𝑢(𝑡,𝑥)𝑡+𝑏𝑡(𝑤,𝑢,𝑢)+𝑏𝑡(𝑢,𝑤,𝑢).(4.4) Here, 𝑏𝑡(𝑣,𝑢,𝑤)=(𝑣𝑖(𝑡,𝑥)𝜕𝑢(𝑡,𝑥)/𝜕𝑥𝑖,𝑤(𝑡,𝑥))𝑡 for 𝑣,𝑢,𝑤𝑊12(Ω𝑡). Since div𝑤=0, then 𝐵𝑡(𝑤,𝑢,𝑢)=Ω𝑡𝜕𝑤𝑖𝑢𝜕𝑥𝑖𝜕𝑤𝑢𝑑𝑥=𝑖𝑢𝜕𝑥𝑖,𝑢𝑡.(4.5) Here, 𝑏𝑡(𝑣,𝑢,𝑤)=(𝑣𝑖(𝑡,𝑥)𝜕𝑢(𝑡,𝑥)/𝜕𝑥𝑖,𝑤(𝑡,𝑥))𝑡 for 𝑣,𝑢,𝑤𝑊12(Ω𝑡). Since div𝑤=0, then 𝐵𝑡(𝑤,𝑢,𝑢)=Ω𝑡𝜕𝑤𝑖𝑢𝜕𝑥𝑖𝜕𝑤𝑢𝑑𝑥=𝑖𝑢𝜕𝑥𝑖,𝑢𝑡.(4.6) Using the inequality |(𝑢,𝑣)𝑡||𝑢|1,𝑡|𝑣|1,𝑡 (see [7, page  252]) and the boundedness of the operator 𝜕/𝜕𝑥𝑖𝑊12(Ω𝑡)𝐿2(Ω𝑡), we have ||𝑏𝑡||||||𝜕𝑤(𝑤,𝑢,𝑢)𝑀𝑖𝑢𝜕𝑥𝑖||||1,𝑡|𝑢|1,𝑡𝑀|𝑤𝑢|0,𝑡|𝑢|1,𝑡.(4.7) With the help of Hölder’s inequality and 𝑣𝐿4(Ω𝑡)𝑀|𝑣|1/21,𝑡|𝑣|1/20,𝑡,𝑣𝑊12Ω𝑡,𝑛=2,(4.8) (see [6, page  233]), we obtain |𝑤𝑢|0,𝑡𝑀𝑤𝐿4(Ω𝑡)𝑢𝐿4(Ω𝑡)𝑀𝑤𝐿4(Ω𝑡)|𝑢|1/21,𝑡|𝑢|1/20,𝑡.(4.9) Using (4.7)–(4.9) and the standard techniques, we obtain, for𝜀>0, ||𝑏𝑡||(𝑤,𝑢,𝑢)𝑀𝑤𝐿4(Ω𝑡)|𝑢|3/21,𝑡|𝑢|1/20,𝑡𝜀|𝑢|21,𝑡+𝑀2(𝜀)𝑤4𝐿4Ω𝑡|𝑢|20,𝑡.(4.10) A similar estimate holds for 𝑏𝑡(𝑢,𝑤,𝑢). Integrating (4.4), using the inequality |(𝐹,𝑢)𝑡||𝐹|1,𝑡|𝑢|1,𝑡, (4.10), and simple calculuses, we have ||||𝑢(𝑡,𝑥)20,𝑡+𝑡0||||𝑢(𝑠,𝑥)2𝑎,𝑠𝑑𝑠𝑀3(𝜀)𝑡0||||𝐹(𝑠,𝑥)21,𝑠𝑑𝑠+𝑀4||||𝑢(0,𝑥)20,0+𝑡0𝑠0||𝑢𝑥||(𝜏,𝑧(𝜏;𝑠,𝑥))0,𝑠||𝑢𝑑𝜏𝑥||(𝑠,𝑥)0,𝑠𝑑𝑠+𝜀𝑡0||||𝑢(𝑠,𝑥))0,𝑠||||𝑢(𝑠,𝑥))1,𝑠𝑑𝑠+𝑀5(𝜀)𝑡0𝑤(𝑠,𝑥)4𝐿4Ω𝑡||||𝑢(𝑠,𝑥))20,𝑠𝑑𝑠.(4.11)
The change of variable 𝑦=𝑧(𝜏;𝑠,𝑥) maps Ω𝑠 into Ω𝜏, by this det𝑧𝑥(𝜏;𝑠,𝑥)1 in virtue of div𝑤=0. Then, ||𝑢𝑥||(𝜏,𝑧(𝜏;𝑠,𝑥)20,𝑠=Ω𝑠||𝑢𝑥||(𝜏,𝑧(𝜏;𝑠,𝑥)2||𝑢𝑑𝑥=𝑥(||𝑠,𝑦)0,𝜏.(4.12) Believing 𝜀 sufficiently small and using the above inequality and (4.11), one easily obtains ||||𝑢(𝑡,𝑥)20,𝑡+𝑡0||||𝑢(𝑠,𝑥)21,𝑠𝑑𝑠𝑀6(𝐹𝑠,𝑥)20,1+||𝑢0||(𝑥)21,0+𝑡0𝑠0||||𝑢(𝜏,𝑥))21,𝜏+𝑑𝜏𝑑𝑠𝑡0𝑤(𝑠,𝑥)4𝐿4(Ω𝑠)||||𝑢(𝑠,𝑥)20,𝑠.𝑑𝑠(4.13) Letting 𝑎(𝑠)=1+𝑤(𝑠,𝑥)4𝐿4Ω𝑠,𝐶1=𝐹20,1+||𝑢0||21,0,(4.14) we obtain from this that, for 𝑏(𝑡)=|𝑢(𝑡,𝑥)|20,𝑡+𝑡0|𝑢(𝑠,𝑥)|2𝑎,𝑠𝑑𝑠, the inequality 𝑏(𝑡)𝑀6𝐶1+𝑡0𝑎(𝑠)𝑏(𝑠)𝑑𝑠(4.15) holds. By Gronwall’s lemma, this yields the inequality 𝑏(𝑡)𝑀6𝐶1𝑀exp6𝑇0𝑎(𝑠)𝑑𝑠𝑀7𝐶1𝑀exp6𝑇0𝑤(𝑠,𝑥)4𝐿4Ω𝑠𝑑𝑠.(4.16) Since (see [13, page  143]) 𝑣𝐿4(𝑄𝑇)𝑀𝑣1,2,𝑣𝑊21,2𝑄𝑇,𝑛=2,(4.17) then 𝑇0𝑤(𝑠,𝑥)4𝐿4Ω𝑠𝑑𝑠𝑀𝑤4𝐿4Ω𝑠𝑀𝑤41,2.(4.18) This and (4.16) yield the inequality 𝑏(𝑡)𝑀8(𝑤1,2)𝐶1, that is, ||||𝑢(𝑡,𝑥)20,𝑡+𝑡0||||𝑢(𝑠,𝑥)2𝑎,𝑠𝑑𝑠𝑀8𝑤1,2(𝐹𝑡,𝑥)0,1+||𝑢0(||𝑥)0,0.(4.19) This gives (4.1).
Theorem 4.1 is proved.

5. Second A Priori Estimate

Theorem 5.1. A solution to the problem (2.3)-(2.4), (2.1) satisfies the inequality sup𝑡||||𝑢(𝑡,𝑥)1,𝑡+𝑢(𝑡,𝑥)1,2𝑀9𝑤1,2,𝐹0,𝑢01,0.(5.1)

To prove Theorem 5.1, we need some estimates for solutions to the more simple linear problem 𝜕𝑢+𝑤𝜕𝑡𝑖𝜕𝑢𝜕𝑥𝑖+𝑢𝑖𝜕𝑤𝜕𝑥𝑖Δ𝑢+=Ψ(𝑡,𝑥),div𝑢(𝑡,𝑥)=0,𝑢(0,𝑥)=𝑢0(𝑥),𝑥Ω0,𝑢(𝑡,𝑥)=0,(𝑡,𝑥)𝑆𝑇.(5.2) In [4], it was found that by a smooth evolution Ω𝑡, 𝜇=0,𝑛=2 for 𝐹𝐿2(𝑄𝑇), 𝑢0𝑉0, problem (2.3)-(2.4) has a unique solution 𝑢𝑊21,2(𝑄𝑇), and the estimate 𝑢1,2𝑀𝑤1,2,𝐹0,||𝑢0||0(5.3) holds. It follows from [4] that the similar result for more simple linear problem (5.2) holds, by this in our case, the estimate 𝑢1,2𝑀𝑤1,2𝜓0+||𝑢0||0(5.4) holds.

It follows from [14] that for functions ̃𝑢𝑊21,2(𝑄) which vanish at 𝑡=0 and on 𝑆𝑇 in a cylindrical domain 𝑄=[0,𝑇]×Ω0, the inequality sup𝑡||||̃𝑢(𝑡,𝑥)1,𝑡𝑀̃𝑢𝑊21,2(𝑄)+||||̃𝑢(0,𝑥)𝑊12(Ω0)(5.5) is valid. The same inequality holds and for 𝑢𝑊21,2(𝑄𝑇). In fact, let ̃𝑢(𝑡,𝑦)=𝑢(𝑡,(̃𝑧(𝑡;0,𝑦)). Since ̃𝑧(𝜏;𝑡,𝑥)) is smooth, then ̃𝑢𝑊21,2(𝑄), and hence inequality (5.5) for ̃𝑢 is valid. By means of substitution 𝑦=̃𝑧(0;𝑡,𝑥), we get the inequality sup𝑡||||𝑢(𝑡,𝑥)1,𝑡𝑀𝑢1,2+||||𝑢(0,𝑥)0,𝑡,𝑢𝑊21,2𝑄𝑇,(5.6) for the original 𝑢(𝑡,𝑥).

Lemma 5.2. Let Ψ𝐿2(𝑄𝑇),𝑢0𝑉0. Then, problem (5.2) has a unique solution, and the estimates sup𝑡||||𝑢(𝑡,𝑥)1,𝑡+𝑢1,2𝑀10𝑤1,2Ψ0+||𝑢0||1,0,(5.7)sup𝑡||𝑢||(𝑡,𝑥)0,𝑡+𝑢(𝑡,𝑥)0,1𝑀11𝑤1,2Ψ0,1+||𝑢0||0,0(5.8) hold.

The proof of (5.7) follows from [4, 14] and (5.6), and (5.8). The proof of (4.1) is similar to one of Theorem 4.1.

Proof of Theorem 5.1. Since the solution 𝑢𝑊21,2(𝑄𝑇), then 𝑢𝑊21,2(𝑄𝑡),0<𝑡𝑇. We remove the second, sixth, and seventh terms on the left-hand side of (2.3) to the right and apply the estimate (5.7) for 𝑄𝑡: 𝑡0|||𝜕𝑢(𝑠,𝑥)|||𝜕𝑠20,𝑠||||𝑑𝑠+𝑢(𝑡,𝑥)20,𝑡+||||𝑢(𝑡,𝑥)21,𝑡+𝑡0||||𝑢(𝑠,𝑥)22,𝑠𝑑𝑠𝑀𝐹20+||𝑈0||21,0+𝑀𝑡0||||𝑢𝑖(𝑠,𝑥)𝜕𝑢(𝑠,𝑥)𝜕𝑥𝑖||||20,𝑠+𝑑𝑠𝑡0||||Div𝑠0||||(𝑢)(𝜏,𝑧(𝜏;𝑠,𝑥))𝑑𝜏20,𝑠𝑑𝑠+𝑡0||||Div𝑠0||||(𝑤)(𝜏,𝑧(𝜏;𝑠,𝑥))𝑑𝜏20,𝑠+𝑑𝑠𝑡0||||𝑤𝑖(𝑡,𝑥)𝜕𝑢(𝑠,𝑥)𝜕𝑥𝑖||||20,𝑠𝑑𝑠+𝑡0||||𝑢𝑖(𝑡,𝑥)𝜕𝑤(𝑠,𝑥)𝜕𝑥𝑖||||20,𝑠𝑑𝑠𝑀𝐶0+𝑀5𝑖=1𝑍𝑖,𝐶0=𝐹20+||𝑢0||21,0.(5.9) Let us estimate 𝑍𝑖. Using Hölder’s inequality and (4.8), we obtain ||||𝑢𝑖(𝑠,𝑥)𝜕𝑢(𝑠,𝑥)𝜕𝑥𝑖||||20,𝑠||||𝑀𝑢(𝑠,𝑥)0,𝑠||||𝑢(𝑡,𝑥)2𝑎,𝑠||||𝑢(𝑠,𝑥)2,𝑠.(5.10) It follows from this and from the uniform boundedness of |𝑢(𝑠,𝑥)|0,𝑠 (in virtue of (4.1)) that ||||𝑢𝑖(𝑠,𝑥)𝜕𝑢(𝑠,𝑥)𝜕𝑥𝑖||||20,𝑠||||𝜀𝑢(𝑠,𝑥)22,𝑠+𝑀𝜀,𝐶0||||𝑢(𝑡,𝑥)4𝑎,𝑠.(5.11) It follows that 𝑍12𝜀𝑡0||||𝑢(𝑠,𝑥)22,𝑠𝑑𝑠+𝑀𝜀,𝐶0𝑡0||||𝑢(𝑠,𝑥)4𝑎,𝑠𝑑𝑠.(5.12) Differentiating, we find that (𝑢=(𝑢1,𝑢2),𝑧=(𝑧1,𝑧2))𝜕𝜕𝑥𝑘𝑠0𝜕𝜕𝑥𝑗𝑢𝑖(𝜏,𝑧(𝜏;𝑠,𝑥))𝑑𝜏=𝑠0𝜕2𝜕𝑥𝑙𝜕𝑥𝑗𝑢𝑖𝜕(𝜏,𝑧(𝜏;𝑠,𝑥))𝜕𝑥𝑘𝑧𝑙(𝜏;𝑠,𝑥)𝑑𝜏.(5.13) It follows that 𝑍22𝑡0𝑠0||𝑢𝑥𝑥||(𝜏,𝑧(𝜏;𝑠,𝑥))0,𝑠𝑧𝑥(𝜏;𝑠,𝑥)𝐶(Ω𝑠)𝑑𝜏2𝑑𝑠.(5.14) Using the change 𝑦=𝑧(𝜏;𝑠,𝑥) in the first factor, estimates (3.3) and (4.1) in the second one, we have 𝑍22𝑀𝑡0𝑠0|𝑢𝑥𝑥|(𝜏,𝑥)0,𝜏𝑑𝜏2𝑑𝑠𝑀𝑡0𝑠0||||𝑢(𝜏,𝑥))22,𝜏𝑑𝜏𝑑𝑠𝑀.(5.15) From here and to the end of the proof, constants 𝑀 depend on 𝐶0 and 𝑤1,2. The estimate of 𝑍3𝑀 is established similarly. The estimates of 𝑍𝑖𝑀,𝑖=4,5, are established more simple.
Estimates of 𝑍𝑖 and (5.9) for small 𝜀>0 yield 𝑡0|||𝜕𝑢(𝑠,𝑥)|||𝜕𝑠20,𝑠||||𝑑𝑠+𝑢(𝑡,𝑥)21,𝑡+𝑡0||||𝑢(𝑠,𝑥)22,𝑠𝑑𝑠𝑀1+𝑡0||||𝑢(𝑠,𝑥)2𝑎,𝑠||||𝑢(𝑠,𝑥)2𝑎,𝑠𝑑𝑠+𝑡0𝑠0||||𝑢(𝜏,𝑥)22,𝑠.𝑑𝜏𝑑𝑠(5.16) It follows from this that for ||||𝑔(𝑡)=𝑢(𝑡,𝑥)21,𝑡+𝑡0||||𝑢(𝑠,𝑥)22,𝑠𝑑𝑠,(5.17) the inequality 𝑔(𝑡)𝑀12+𝑀13𝑡0||||𝑢(𝑠,𝑥)2𝑎,𝑠𝑔(𝑠)𝑑𝑠(5.18) holds. Consequently, 𝑔(𝑡)𝑀12exp𝑡0𝑀13||||𝑢(𝑠,𝑥)2𝑎,𝑠𝑑𝑠.(5.19) This and (4.1) give that 𝑔(𝑡)𝑀14𝑤1,2,𝐶0.(5.20) But then the right-hand side of (5.16) is bounded uniformly with respect to 𝑡. Hence, the left-hand side is bounded also.
Inequality (5.1) is proved.
Theorem 5.1 is proved.

6. Reduction to Operator Equation

Let 𝑢=(Ψ) be a solution to (5.2) by 𝑢0=0 and given Ψ𝐿2(𝑄𝑇). Obviously, the expression 𝑢=(Ψ) defines a bounded linear operator from 𝐿2(𝑄𝑇) to 𝑊21,2(𝑄𝑇). Using the operator , (2.5), and Lemma 5.2, we rewrite the problem (2.3)-(2.4), (2.1) by 𝑢0= 0 as 𝑢𝑢=(𝐹)𝑖𝜕𝑢𝜕𝑥𝑖+𝜇Div𝑡0(𝑢)(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠+𝜇Div𝑡0(𝑤)(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠=Φ+𝑆1(𝑢)+𝜇𝑆2(𝑢)+𝜇𝑆3(𝑢)𝑄(𝑢).(6.1) Thus, the problem (1.1)-(1.2), (2.1) with 𝑢0=0 is reduced to equation 𝑢=𝑄(𝑢).(6.2) We show that for sufficiently large 𝑅 and sufficiently small 𝑇>0, operator equation (6.2) is uniquely solvable in 𝑆(𝑅)=𝑢𝑢𝑊21,2𝑄𝑇,𝑢(𝑡,𝑥)𝑉𝑡,𝑢1,2𝑅,𝑢(0,𝑥)=0.(6.3)

7. Solvability of (6.2) for Small 𝑇

Lemma 7.1. By sufficiently large 𝑅 and sufficiently small 𝑇, the operator 𝑄 maps 𝑆(𝑅) into itself.

Proof. Let 𝑢𝑆(𝑅). Properties of the operator and (6.1) imply 𝑢(0,𝑥)=0 and 𝑄(𝑢)1,2Φ0+𝑆10𝑆+𝜇20𝑆+𝜇30.(7.1) Let us show that 𝑆𝑖0𝑀(𝑅)𝑇1/2,𝑖=1,2,3.(7.2) From (5.10), it follows that ||𝑆1||0,𝑡𝑀|𝑢|1/20,𝑡|𝑢|1,𝑡|𝑢|1/22,𝑡.(7.3) Hence, from (5.6), the Hölder inequality, and 𝑢1,2𝑅, it follows that 𝑆120𝑀𝑇0||𝑆1||(𝑡,𝑥)20,𝑡𝑑𝑡𝑀sup𝑡||||𝑢(𝑡,𝑥)0,𝑡sup𝑡||||𝑢(𝑡,𝑥)21,𝑡𝑇0||||𝑢(𝑡,𝑥)2,𝑡𝑑𝑡𝑀𝑅3𝑇0||||𝑢(𝑡,𝑥)2,𝑡𝑑𝑡𝑀𝑅3𝑢1,2𝑇1/2𝑀𝑅4𝑇1/2.(7.4) Estimate 𝑆20. Differentiating and applying the integral Minkowski inequality, we have ||𝑆2||0,𝑡𝑀𝑇0||𝑢𝑥||(𝑠,𝑧(𝑠;𝑡,𝑥)0,𝑡𝑧𝑥(𝑠;𝑡,𝑥)𝐶(Ω𝑡)𝑑𝑠.(7.5) Making the change 𝑦=𝑧(𝑠;𝑡,𝑥), we get ||𝑢𝑥||(𝑠,𝑧(𝑠;𝑡,𝑥)0,𝑡=||𝑢𝑥||(𝑠,𝑦)0,𝑠.(7.6) From (3.3), it follows that 𝑧𝑥(𝑠;𝑡,𝑥)𝐶(Ω𝑡)𝑀||||𝑀exp𝜏0||||𝑣(𝑠,𝑥)𝑎,𝑠||||𝑑𝑠𝑀exp𝑀(𝑅).(7.7) From this by means of Hölder inequality, it follows that ||𝑆2||0,𝑡𝑀(𝑅)𝑇0||𝑢𝑥||(𝑠,𝑦)0,𝑠𝑑𝑠𝑀(𝑅)𝑇1/2𝑢0,1𝑀(𝑅)𝑇1/2.(7.8) Integrating, we obtain (7.2) for 𝑖=2. Estimate (7.2) for 𝑖=3 is established easier. The statement of Lemma 7.1 follows from estimates (7.1) and (7.2).
Lemma 7.1 is proved.

Now consider the 𝑆(𝑅) as a metric space with the generated by the norm 0,1 metric. From the definitions of generalized derivatives and the weak compactness of a ball in 𝑊21,2(𝑄𝑇), it follows that the metric space 𝑆(𝑅) is complete ([10, Chapter  3, Section  1]).

Lemma 7.2. By sufficiently small 𝑇, the operator 𝑄 on 𝑆(𝑅) is a contraction in the metric 0,1.

Proof. Let 𝑢1,𝑢2𝑆(𝑅). Then, 𝑢Δ𝑄(𝑢)=𝑄2𝑢𝑄1𝜕=𝜕𝑥𝑖𝑢1𝑖𝑢1𝑢2𝑖𝑢2+𝜇Div𝑡0𝑢2𝑠,𝑧2𝑢(𝑠;𝑡,𝑥)1𝑠,𝑧1(𝑠;𝑡,𝑥)𝑑𝑠+𝜇Div𝑡0(𝑤)𝑠,𝑧2(𝑠;𝑡,𝑥)(𝑤)𝑠,𝑧1(𝑠;𝑡,𝑥)𝑑𝑠=𝑍1+𝜇𝑍2+𝜇𝑍3.(7.9) From the inequality |𝜕𝑣/𝜕𝑥𝑖|1,𝑡𝑀|𝑉|0,𝑡 and (5.8), it follows that 𝑍120,1𝑍𝑀120,1=𝑀𝑇0||||𝜕𝜕𝑥𝑖𝑢2𝑖𝑢2𝑢1𝑖𝑢1||||21,𝑡𝑑𝑡𝑀𝑇02𝑖=1||𝑢2𝑖𝑢2𝑢1𝑖𝑢𝑎||20,𝑡𝑑𝑡.(7.10) By means of simple calculuses, we obtain ||𝑢𝑔=2𝑖𝑢2𝑢1𝑖𝑢𝑎||20,𝑡Ω𝑡||𝑢2𝑖𝑢1𝑖||2||𝑢𝑎||2𝑑𝑥+Ω𝑡||𝑢1𝑖||2||𝑢2𝑢𝑎||2𝑢𝑑𝑥𝑀2𝑢12𝐿4Ω𝑡𝑢22𝐿4Ω𝑡+𝑢12𝐿4Ω𝑡.(7.11) It follows from the continuous imbedding 𝑊12(Ω𝑡)𝐿4(Ω𝑡), (4.8), and (5.6) that ||𝑢𝑔𝑀2𝑢1||21,𝑡max𝑘=1,2||𝑢𝑘||1,𝑡||𝑢𝑘||0,𝑡||𝑢𝑀𝑅2𝑢1||21,𝑡max𝑘=1,2||𝑢𝑘||0,𝑡.(7.12) The inequality (4.2) for 𝜓=|𝑢𝑖(𝑡,𝑥)|2 and (4.3) by 𝑢𝑖(𝑡,𝑥)=0 on 𝑆𝑇 give 𝜕||𝑢𝑖||(𝑡,𝑥)20,𝑡𝑑𝑡=2𝜕𝑢𝑖(𝑡,𝑥)𝜕𝑡,𝑢𝑖(𝑡,𝑥)𝑡.(7.13) Hence, from 𝑢𝑖(0,𝑥)=0 and (5.6), we obtain ||𝑢𝑖(||𝑡,𝑥)20,𝑡2𝑡0|||𝜕𝑢𝜕𝑠𝑖(|||𝑠,𝑥)0,𝑠||𝑢𝑖(||𝑠,𝑥)0,𝑠𝑑𝑠𝑀sup𝑡||𝑢𝑖||(𝑡,𝑥)0,𝑡𝑡0|||𝜕𝑢𝜕𝑠𝑖|||(𝑠,𝑥)0,𝑠𝑑𝑠𝑀𝑅2𝑇1/2.(7.14) It follows from (7.11)–(7.14) that 𝐺𝑀(𝑅)𝑇1/2|𝑢2𝑢𝑎|21,𝑡. This and (7.10) yield 𝑍120,1𝑀(𝑅)𝑇1/2𝑇0||𝑢2𝑢1||21,𝑠𝑑𝑠𝑀(𝑅)𝑇1/2𝑢2𝑢120,1.(7.15) Let us estimate 𝑍2. As by the evaluation of 𝑍1, we have 𝑍20,1𝑍𝑀20,1𝑀𝑡0𝑢2𝑠,𝑧2(𝑢𝑠;𝑡,𝑥)1𝑠,𝑧1(𝑠;𝑡,𝑥)𝑑𝑠0.(7.16) Denoting the expression under the sign of 0 by 𝐺, we get 𝐺=𝑡0𝑢2𝑠,𝑧2(𝑢𝑠;𝑡,𝑥)2𝑠,𝑧1(+𝑠;𝑡,𝑥)𝑑𝑠𝑡0𝑢2𝑠,𝑧1𝑢(𝑠;𝑡,𝑥)1𝑠,𝑧1𝐺(𝑠;𝑡,𝑥)𝑑𝑠=1+𝐺2.(7.17) Let us estimate the first term. As in [10, Chapter  3, Section  1.2], we continue the 𝑢1 and 𝑢2 out of 𝑄𝑇 on 𝑅𝑛 with the preservation of class so that the velocity fields 𝑣1=𝑢1+𝑤 and 𝑣2=𝑢2+𝑤 of the corresponding Cauchy problems (2.1) are vanished out of a compact convex domain Ω, ΩΩ𝑡,0𝑡𝑇. By the uniqueness theorem for Cauchy problems (7.11), solutions 𝑧1 and 𝑧2 are the restrictions of the corresponding extended Cauchy problems. We mark the extended 𝑢1, 𝑢2, 𝑧1, and 𝑧2 by the bar at the top.
We now estimate 𝐺1. Using the Newton-Leibnitz formula, we have 𝐵𝑖=𝜕𝜕𝑥𝑖𝑢2𝑠,𝑧2𝜕(𝑠;𝑡,𝑥)𝜕𝑥𝑖𝑢2𝑠,𝑧1=(𝑠;𝑡,𝑥)2𝑘=110𝜕2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑠,𝑦(𝑠,𝑡,𝑥,𝛼)𝑑𝛼𝑧1𝑘(𝑠;𝑡,𝑥)𝑧2𝑘,(𝑠;𝑡,𝑥)𝑧=𝑧1,𝑧2.(7.18) Here, 𝑌=𝑦(𝑠,𝑡,𝑥,𝛼)=𝛼𝑧1(𝑠;𝑡,𝑥)+(1𝛼)𝑧2(𝑠;𝑡,𝑥).
As in [10, Chapter  3, Section  1.2] in the cylindrical case 𝑄=[0,𝑇]×Ω, we demonstrate that, for small 𝑇 and 0𝛼1, the mapping 𝑦(𝑠,𝑡,𝑥,𝛼) is diffeomorphism of Ω to Ω, and the Jacobians of the direct and inverse mappings are uniformly bounded. It follows that 𝐵𝑖𝐿2(Ω𝑡)𝑏𝑖𝐿2(Ω)𝑀2𝑘=110𝜕2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑠,𝑦(𝑠,𝑡,𝑥,𝛼)𝐿2(Ω)×𝑑𝛼𝑧1(𝑠;𝑡,𝑥)𝑧2(𝑠;𝑡,𝑥)𝐶(Ω).(7.19) Using the change of variable 𝑦=𝑦(𝑠,𝑡,𝑥,𝛼), we get 𝜕2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑠,𝑦(𝑠,𝑡,𝑥,𝛼)𝐿2(Ω)𝜕𝑀2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑗(𝑠,𝑦)𝐿2(Ω)𝜕𝑀2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑗(𝑠,𝑦)𝐿2(Ω𝑠)||𝑢𝑀2||(𝑠,𝑦)2,𝑠.(7.20) Using (7.20), we have 𝜕2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑗(𝑠,𝑦(𝑠,𝑡,𝑥,𝛼)𝐿2(Ω𝑡)𝜕𝑀2𝜕𝑥𝑘𝜕𝑥𝑖𝑢2𝑠,𝑦(𝑠,𝑡,𝑥,𝛼)𝐿2(Ω)||𝑢𝑀2||(𝑠,𝑦)2,𝑠.(7.21) From this estimate, (3.3), and (3.6), it follows that ||𝐺1||0,𝑡𝑀𝑡0||𝑢2(||𝑠,𝑦)2,𝑠𝑧1(𝑠;𝑡,𝑥)𝑧2(𝑠;𝑡,𝑥)𝐶(Ω𝑡)𝑑𝑠𝑀(𝑅)𝑇1/2𝑢20,2𝑢2𝑢10,1𝑀(𝑅)𝑇1/2𝑢2𝑢10,1.(7.22) A similar estimate for 𝐺2 is established easier. From the estimates (7.16), (7.21), (7.22), the inequality 𝑍20,1𝑀(𝑅)𝑇1/2𝑢2𝑢10,1(7.23) follows.
A similar estimate for 𝑍30,1 is established similarly. From the estimates 𝑍𝑖0,1 for 𝑖=1,2,3, it follows the inequality 𝑄𝑢2𝑢𝑄10,1𝑀(𝑅)𝑇1/4𝑢2𝑢10,1𝑢𝑞2𝑢10,1,(7.24) where 𝑞(0,1) by sufficiently small 𝑇.
Lemma 7.2 is proved.

Lemma 7.3. For sufficiently large 𝑅 and sufficiently small 𝑇, the operator 𝑄 in 𝑆(𝑅) has a unique fixed point.

Proof. The proof follows from Lemmas 7.2 and 7.1 by the principle of contracting maps. Lemma 7.3 is proved.

Proof of Theorem 2.2. The theorem for a small 𝑇 and 𝑢0=0 follows from Lemma 7.3 by the equivalence of problem (2.3)-(2.4), (2.1), and (6.2). Establish the solvability of the problem (2.3)-(2.4), (2.1) at small 𝑇 and 𝑢00.Let 𝑢0(𝑡,𝑥) be a solution to the linear problem (5.2) by 𝑢00 and 𝐹=0. There exists only one solution to this problem in the force of Lemma 5.2, and the inequality sup𝑡||𝑢0||(𝑡,𝑥)1,𝑡+𝑢01,2𝑀11𝑤1,2||𝑢0||1,0(7.25) holds.
Represent the solution to (2.3)-(2.4), (2.1) as 𝑢(𝑡,𝑥)=𝑢(𝑡,𝑥)+𝑢0(𝑡,𝑥). It is easy to see that 𝑢(𝑡,𝑥) satisfies relationships 𝜕𝑢+𝜕𝑡𝑢𝑖𝜕𝑢𝜕𝑥𝑖+𝑤𝑖𝜕𝑢𝜕𝑥𝑖+𝑢𝑖𝜕𝑤𝜕𝑥𝑖Δ𝑢𝜇Div𝑡0𝑢(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠𝜇Div𝑡0𝑢0(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠𝜇Div𝑡0𝑤(𝑠,𝑧(𝑠;𝑡,𝑥))𝑑𝑠+grad𝑝(𝑡,𝑥)=𝐹(𝑡,𝑥),(𝑡,𝑥)𝑄𝑇,div𝑢(𝑡,𝑥)=0,(𝑡,𝑥)𝑄𝑇,𝑢(0,𝑥)=0,𝑥Ω0,𝑢(𝑡,𝑥)=0,(𝑡,𝑥)𝑆𝑇,(7.26) and (2.1). Here, 𝑤=𝑤+𝑢0, 𝐹 is defined as in (2.3)-(2.4), (2.1), 𝜕𝐹(𝑡,𝑥)=𝐹(𝑡,𝑥)𝑢0𝜕𝑡𝑢0𝑖𝜕𝑢0𝜕𝑥𝑖𝑤𝑖𝜕𝑢0𝜕𝑥𝑖𝑢0𝑖𝜕𝑤𝜕𝑥𝑖+Δ𝑢0,(7.27) and 𝑧 is a solution to problem (2.1) by 𝑣=𝑢+𝑤+𝑢0. In this case, 𝐹,𝑤𝐿2(𝑄𝑇). In fact, by 𝑔,𝑣𝐿4(𝑄𝑇) with the help of Hölder’s inequality, we have 𝑔𝑣0𝑔𝐿4(𝑄𝑇)𝑣𝐿4(𝑄𝑇).(7.28) It follows from this and (4.17) that by 𝑔,𝑣𝑊21,2(𝑄𝑇), the inequality 𝑔𝑣0𝑀𝑔1,2𝑣1,2(7.29) holds. Setting 𝑔=𝑤,𝑣=𝑢0, we obtain the required inclusion. Note that this and (7.25) imply the estimates 𝐹0𝑀12𝑤1,2,𝐹1,2,𝑢01,2,𝑤1,2𝑀13𝑤1,2+𝑢01,2.(7.30) Thus, we have reduced the case of nonzero initial condition to the case considered of zero initial conditions. It follows from the above that for small 𝑇=𝑇(𝐹0,𝑤1,2), there exists a unique solution 𝑢 to problem (7.26), (2.1), and the inequality sup𝑡||||𝑢(𝑡,𝑥)1,𝑡+𝑢1,2𝑀14𝐹0,𝑤1,2(7.31) holds.
The inequalities (7.30)–(7.32) imply that sup𝑡||||𝑢(𝑡,𝑥)1,𝑡+𝑢1,2𝑀15𝐹1,2,𝑤1,2,||𝑢0||1,0.(7.32) Finally, the unique solvability of (7.26), (2.1) on [0,𝑇] at small 𝑇 implies the existence and uniqueness on [0,𝑇] of the solution 𝑢 to problem (2.3)-(2.4), (2.1) by 𝑢00 and the inequality sup𝑡||||𝑢(𝑡,𝑥)1,𝑡+𝑢1,2𝑀16𝐹1,2,𝑤1,2,||𝑢0||1,0.(7.33) Now let 𝑇>0 be arbitrary. Let 𝑇0 such that our problem is uniquely solvable on [0,𝑇0][0,𝑇]. Note that the same result holds not only on [0,𝑇0], but also on [𝑡,𝑡+𝑇0][0,𝑇] for any 𝑡>0. Let 𝑇𝑘=(𝑘𝑇)/𝑁, 𝑘=1,2,,𝑁, 𝑁 be a natural number. Consider problem (2.3)-(2.4), (2.1) on the domains 𝑄𝑘, where 𝑄𝑘={(𝑡,𝑥)𝑡𝑘𝑡𝑡𝑘+1,𝑥Ω𝑡}. Let us find solutions 𝑢𝑘(𝑡,𝑥) to problem (2.3)-(2.4), (2.1) from 𝑊21,2(𝑄𝑘), replacing by 𝑘>1 the initial condition 𝑢(0,𝑥)=𝑢0(𝑥) by the initial conditions 𝑢𝑘(𝑡𝑘,𝑥)=𝑢𝑘1(𝑡𝑘,𝑥), 𝑥Ω𝑘. From the local solvability, which is established above, it follows that the length of the segment [𝑇𝑘,𝑡𝑘+1], at which these problems are uniquely solvable are determined by 𝐹𝐿2(𝑄𝑘) and |𝑈𝑘1(𝑡𝑘,𝑥)|1,𝑡𝑘. If these values are uniformly bounded with respect to 𝑡, then there exists a sufficiently large 𝑁, that all these problems are uniquely solvable at [𝑡𝑘,𝑡𝑘+1]. We solve these problems sequentially, starting with 𝑘=0. By a priori estimate (5.1), |𝑢𝑘1(𝑡𝑘,𝑥)|1,𝑡𝑘 is uniformly bounded on 𝑡. In addition, it is obvious that 𝐹𝐿2(𝑄𝑘)𝐹𝐿2(𝑄𝑇). Thus, we obtain the solvability to problems for all 𝑘. The function 𝑢(𝑡,𝑥), whose restriction to [𝑡𝑘,𝑡𝑘+1] is 𝑢𝑘(𝑡,𝑥), is obviously the unique solution to (2.3)-(2.4), (2.1) at 𝑄𝑇. Theorem 2.2 is proved completely.

Proof of Theorem 2.1. The conditions of Theorem 2.1 and (2.5) imply that the following conditions of Theorem 2.2 are fulfilled. Obviously, the function 𝑣=𝑢+𝑤, where 𝑢 is a solution to (2.3)-(2.4), (2.1), is the unique solution to (1.1)-(1.2), (2.1). Theorem 2.1 is proved.

Acknowledgments

The research was partially supported by Russian Foundation for Basic Researches, Grant 04-01-0008.

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