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 , , be a family of bounded domains with , depending smoothly on . Consider the initial-boundary problem where , .

In (1.1)-(1.2), the velocity and the pressure are unknown, is given outer forces, is prescribed initial velocity, = is the rate-of-strain tensor, that is, the matrix with elements , . The divergence Div of a matrix is defined as a vector with elements which are the divergences of the lines. The vector function is a solution to the Cauchy problem (in the integral form) The system describes the evolution of a viscoelastic medium with a memory along the trajectories of the velocity vector field.

In the case of a cylindrical domain (), the local existence and uniqueness theorem (e.u.t.) for strong solutions (, ) 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 () to the problem (1.1)-(1.2), (1.3), where in (1.3) is replaced by some regularized vector field (regularized problem) for , was established in [3]. There was given the motivation of the regularization of the velocity field . In the case that and is noncylindrical domain, the local for and nonlocal for 2 (e.u.t.) of strong solutions () 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 in noncylindrical domain was established in [5].

Our goal is to prove a nonlocal e.u.t. for solutions , , by 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 of the pair . The function 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 , sometimes with indices. We take the summation convention for repeated indices.

2. Notations and Statement of the Results

The norms of a function in real spaces , and in Sobolev spaces , we denote by , , , , respectively. We use the same designation for scalar, vector, or matrix functions.

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

We denote by the orthogonal projector in on . Consider in unbounded linear operator , with the domain . Operator (see [8, page  54]) is a positive self-adjoint operator. We define the regularizer , as , where . Consider the Cauchy problem where . If and the evolution of is smooth, then regularized problem (2.1) is uniquely solvable (see below). As to regularization operator , we note that it is uniformly bounded of as an operator from in . There are other constructions of such operators (see, e.g., [3, 9]). Every one of them has the property of strong convergence of to by .

We believe that , where is a solution to the Cauchy problem where is sufficiently smooth on some neighborhood of solenoidal function. The boundary condition (1.2) is defined as , under the natural condition , . Note that the same admits representation via evolution by means of different functions with correspondent boundary values .

The main result reads as follows.

Theorem 2.1. Let , , , . Then, problem (1.1)-(1.2), (2.1) has a unique solution .

As to the form of and the boundary function , we preferred to concentrate ourselves on the essence of the problem and chose a simple way to provide the solvability of (2.1) on and to avoid a complexity in the proofs. Let the assumption be a sufficiently arbitrary domain, and let be the trace of a solenoidal and satisfy ( 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 .

We believe also , and is sufficiently smooth in order to simplify the calculuses (in fact, it is sufficient that .

We reduce the original problem to a problem with zero boundary data. Let , where a solenoidal function and on . Then, satisfies relations and (2.1). Here, , 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 , . Then, problem (2.3)-(2.4), (2.1) has a unique solution .

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 , satisfying (1.3). Denote by a Jacobi matrix of the vector function , by a tensor which is composed of second derivatives of the components of a vector function , and by and the sum of squares of -norms of the first and second derivatives, respectively. The following result was established in [1] for a cylindrical domain .

Lemma 3.1 (see [2]). Let and vanish at . Then, the corresponding Cauchy problems (1.3) are uniquely solvable, and the estimates hold. If, moreover, , then

Using the smoothness of the evolution of and the fact that the lateral side consists of the trajectories of a smooth velocity field , we can show that the similar result holds for noncylindrical domain by on . To do this, one has as in [10, Chapter  3, Section  1.2] to continue on with the preservation of the class by functions vanishing outside some compact domain , 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 , properties of allow to strengthen the result of Lemma 3.1.

Lemma 3.2. Let . Then, corresponding Cauchy problems (2.1) are uniquely solvable, and the estimates hold.

In the proof of Lemma 3.2, Lemma 3.1 and uniform with respect to estimate are used. The last estimate easily follows from the continuous imbedding , (see [11, page  408]), and the inclusion .

4. First A Priori Estimate

Theorem 4.1. Let be a solution to the problem (2.3)-(2.4), (2.1). Then, inequality holds.

Recall that .

Proof. Let be a sufficiently smooth scalar function. Since , where is a solution to the Cauchy problem (2.2) for with , then (see [12, page  8]) It follows that where , , and vanishes on .
Multiply both sides of (2.3) in by . The standard arguments as in [5] and (4.2), (4.3) give Here, for . Since , then Here, for . Since , then Using the inequality (see [7, page  252]) and the boundedness of the operator , we have With the help of Hölder’s inequality and (see [6, page  233]), we obtain Using (4.7)–(4.9) and the standard techniques, we obtain, for, A similar estimate holds for . Integrating (4.4), using the inequality , (4.10), and simple calculuses, we have
The change of variable maps into , by this in virtue of . Then, Believing sufficiently small and using the above inequality and (4.11), one easily obtains Letting we obtain from this that, for , the inequality holds. By Gronwall’s lemma, this yields the inequality Since (see [13, page  143]) then This and (4.16) yield the inequality , that is, 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

To prove Theorem 5.1, we need some estimates for solutions to the more simple linear problem In [4], it was found that by a smooth evolution , for , , problem (2.3)-(2.4) has a unique solution , and the estimate holds. It follows from [4] that the similar result for more simple linear problem (5.2) holds, by this in our case, the estimate holds.

It follows from [14] that for functions which vanish at and on in a cylindrical domain , the inequality is valid. The same inequality holds and for . In fact, let . Since is smooth, then , and hence inequality (5.5) for is valid. By means of substitution , we get the inequality for the original .

Lemma 5.2. Let . Then, problem (5.2) has a unique solution, and the estimates 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 , then . 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 : Let us estimate . Using Hölder’s inequality and (4.8), we obtain It follows from this and from the uniform boundedness of (in virtue of (4.1)) that It follows that Differentiating, we find that It follows that Using the change in the first factor, estimates (3.3) and (4.1) in the second one, we have From here and to the end of the proof, constants depend on and . The estimate of is established similarly. The estimates of , are established more simple.
Estimates of and (5.9) for small yield It follows from this that for the inequality holds. Consequently, This and (4.1) give that But then the right-hand side of (5.16) is bounded uniformly with respect to . Hence, the left-hand side is bounded also.
Inequality (5.1) is proved.
Theorem 5.1 is proved.