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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 407940, 19 pages
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.
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.
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 . 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 . The e.u.t. of weak solutions to the regularized problem (1.1)-(1.2), (1.3) with in noncylindrical domain was established in .
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.
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 ) 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).
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  for a cylindrical domain .
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.
Lemma 3.2. Let . Then, corresponding Cauchy problems (2.1) are uniquely solvable, and the estimates hold.
4. First A Priori Estimate
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  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
To prove Theorem 5.1, we need some estimates for solutions to the more simple linear problem In , 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  that the similar result for more simple linear problem (5.2) holds, by this in our case, the estimate holds.
It follows from  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.
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.
6. Reduction to Operator Equation
Let be a solution to (5.2) by and given . Obviously, the expression defines a bounded linear operator from to . Using the operator , (2.5), and Lemma 5.2, we rewrite the problem (2.3)-(2.4), (2.1) by 0 as Thus, the problem (1.1)-(1.2), (2.1) with is reduced to equation We show that for sufficiently large and sufficiently small , operator equation (6.2) is uniquely solvable in
7. Solvability of (6.2) for Small
Lemma 7.1. By sufficiently large and sufficiently small , the operator maps into itself.
Proof. Let . Properties of the operator and (6.1) imply and
Let us show that
From (5.10), it follows that
Hence, from (5.6), the Hölder inequality, and , it follows that
Estimate . Differentiating and applying the integral Minkowski inequality, we have
Making the change , we get
From (3.3), it follows that
From this by means of Hölder inequality, it follows that
Integrating, we obtain (7.2) for . Estimate (7.2) for 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 as a metric space with the generated by the norm metric. From the definitions of generalized derivatives and the weak compactness of a ball in , it follows that the metric space is complete ([10, Chapter 3, Section 1]).
Lemma 7.2. By sufficiently small , the operator on is a contraction in the metric .
Proof. Let . Then,
From the inequality and (5.8), it follows that
By means of simple calculuses, we obtain
It follows from the continuous imbedding , (4.8), and (5.6) that
The inequality (4.2) for and (4.3) by on give
Hence, from and (5.6), we obtain
It follows from (7.11)–(7.14) that . This and (7.10) yield
Let us estimate . As by the evaluation of , we have
Denoting the expression under the sign of by , we get
Let us estimate the first term. As in [10, Chapter 3, Section 1.2], we continue the and out of on with the preservation of class so that the velocity fields and of the corresponding Cauchy problems (2.1) are vanished out of a compact convex domain , . By the uniqueness theorem for Cauchy problems (7.11), solutions and are the restrictions of the corresponding extended Cauchy problems. We mark the extended , , , and by the bar at the top.
We now estimate . Using the Newton-Leibnitz formula, we have Here, .
As in [10, Chapter 3, Section 1.2] in the cylindrical case , we demonstrate that, for small and , the mapping is diffeomorphism of to , and the Jacobians of the direct and inverse mappings are uniformly bounded. It follows that Using the change of variable , we get Using (7.20), we have From this estimate, (3.3), and (3.6), it follows that A similar estimate for is established easier. From the estimates (7.16), (7.21), (7.22), the inequality follows.
A similar estimate for is established similarly. From the estimates for , it follows the inequality where by sufficiently small .
Lemma 7.2 is proved.
Lemma 7.3. For sufficiently large and sufficiently small , the operator in has a unique fixed point.
Proof of Theorem 2.2. The theorem for a small and 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 and .Let be a solution to the linear problem (5.2) by and . There exists only one solution to this problem in the force of Lemma 5.2, and the inequality
Represent the solution to (2.3)-(2.4), (2.1) as . It is easy to see that satisfies relationships and (2.1). Here, , is defined as in (2.3)-(2.4), (2.1), and is a solution to problem (2.1) by . In this case, . In fact, by with the help of Hölder’s inequality, we have It follows from this and (4.17) that by , the inequality holds. Setting , we obtain the required inclusion. Note that this and (7.25) imply the estimates 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 , there exists a unique solution to problem (7.26), (2.1), and the inequality holds.
The inequalities (7.30)–(7.32) imply that Finally, the unique solvability of (7.26), (2.1) on at small implies the existence and uniqueness on of the solution to problem (2.3)-(2.4), (2.1) by and the inequality Now let be arbitrary. Let such that our problem is uniquely solvable on . Note that the same result holds not only on , but also on for any . Let , , be a natural number. Consider problem (2.3)-(2.4), (2.1) on the domains , where . Let us find solutions to problem (2.3)-(2.4), (2.1) from , replacing by the initial condition by the initial conditions , . From the local solvability, which is established above, it follows that the length of the segment , at which these problems are uniquely solvable are determined by and . If these values are uniformly bounded with respect to , then there exists a sufficiently large , that all these problems are uniquely solvable at . We solve these problems sequentially, starting with . By a priori estimate (5.1), is uniformly bounded on . In addition, it is obvious that . Thus, we obtain the solvability to problems for all . The function , whose restriction to is , is obviously the unique solution to (2.3)-(2.4), (2.1) at . 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 , where 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.
The research was partially supported by Russian Foundation for Basic Researches, Grant 04-01-0008.
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