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

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.


Introduction
Let Ω t ⊂ R n , n 2, 3, be a family of bounded domains with ∂Ω t ∈ C 2 , depending smoothly on 0 ≤ t ≤ T, < ∞.Consider the initial-boundary problem where Q T { t, x : 0 ≤ t ≤ T, x ∈ Ω t }, S T { t, x : t ∈ 0, T , x ∈ ∂Ω t }.In 1.1 -1.2 , the velocity v t, x v 1 t, x , . . ., v n t, x and the pressure p t, x are unknown, f t, x is given outer forces, v 0 x is prescribed initial velocity, E v {E ij } n i,j 1 is the rate-of-strain tensor, that is, the matrix with elements E ij v 1/2 ∂v i /∂x j ∂v j /∂x i , μ 0 > 0, μ ≥ 0, λ ≥ 0. 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 τ t v s, z s; t, x ds, τ ∈ 0, T , t, x ∈ Q T . 1.3 The system describes the evolution Ω t of a viscoelastic medium Ω 0 with a memory along the trajectories of the velocity vector field.
In the case of a cylindrical domain Q T Ω t ≡ Ω 0 , the local existence and uniqueness theorem e.u.t. for strong solutions v ∈ W 1,2 q Q T , 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 v ∈ W 1,2 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 Q T is noncylindrical domain, the local for n 3 and nonlocal for n 2 e.u.t. of strong solutions v ∈ W 1,2 2 Q T 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 Q T 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.

Notations and Statement of the Results
The norms of a function u in real spaces , 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 where Q T and the evolution of Ω t is smooth, then regularized problem 2.1 is uniquely solvable see below .As to regularization operator S δ , we note that it is uniformly bounded of t as an operator from L 2 Ω t in C 1 Ω t .There are other constructions of such operators see, e.g., 3, 9 .Every one of them has the property of strong convergence of S δ to I by δ → 0.
We believe that Ω t z t, 0, Ω 0 , where z τ; t, x is a solution to the Cauchy problem where w is sufficiently smooth on some neighborhood of Q T solenoidal function.The boundary condition 1.2 is defined as ψ t, x w t, x , t, x ∈ S T under the natural condition v 0 x ψ 0, x , x ∈ ∂Ω 0 .Note that the same Q T admits representation via evolution Ω t by means of different functions w with correspondent boundary values ψ.
The main result reads as follows.
As to the form of Q T 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 Q T be a sufficiently arbitrary domain, and let ψ t, x be the trace of a solenoidal ψ t, x ∈ W 1,2 2 Q T 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, μ 0 1, and w is sufficiently smooth in order to simplify the calculuses in fact, it is sufficient that w ∈ W 1,2 q Q , q > n .We reduce the original problem to a problem with zero boundary data.Let v u w, where a solenoidal function w ∈ W The proof of Theorem 2.2 consists of several steps.

Properties of Solutions to Cauchy Problem
By a solution to 1.
and vanish at S T .Then, the corresponding Cauchy problems 1.3 are uniquely solvable, and the estimates Using the smoothness of the evolution of Ω t 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 Q T by v, v 1 , v 2 ψ on S T .To do this, one has as in 10, Chapter 3, Section 1.2 to continue v, v 1 , v 2 on R n with the preservation of the class by functions vanishing outside some compact domain Q 0, T × Ω, Ω ⊃ Ω t , 0 ≤ t ≤ T , 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.
Then, corresponding Cauchy problems 2.1 are uniquely solvable, and the estimates In the proof of Lemma 3.2, Lemma 3.1 and uniform with respect to

First A Priori Estimate
Theorem 4.1.Let u be a solution to the problem 2.3 -2.4 , 2.1 .Then, inequality Proof.Let Ψ t, x be a sufficiently smooth scalar function.Since Ω t z t, 0, Ω 0 , where z is a solution to the Cauchy problem 2.2 for w with div w 0, then see 12, page 8

4.11
The change of variable y z τ; s, x maps Ω s into Ω τ , by this det z x τ; s, x ≡ 1 in virtue of div w 0.Then, Believing ε sufficiently small and using the above inequality and 4.11 , one easily obtains

5.1
To prove Theorem 5.1, we need some estimates for solutions to the more simple linear problem ∂u ∂t

5.2
In 4 , it was found that by a smooth evolution Ω t , μ 0, n 2 for Q which vanish at t 0 and on S T in a cylindrical domain Q 0, T × Ω 0 , the inequality is valid.The same inequality holds and for u ∈ W 1,2 2 Q T .In fact, let u t, y u t, z t; 0, y .Since z τ; t, x is smooth, then u ∈ W 1,2 2 Q , and hence inequality 5.5 for u is valid.By means of substitution y z 0; t, x , we get the inequality for the original u t, x .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.

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

5.15
From here and to the end of the proof, constants M depend on C 0 and w 1,2 .The estimate of Z 3 ≤ M is established similarly.The estimates of Z i ≤ M, i 4, 5, are established more simple.
Estimates of Z i and 5.9 for small ε > 0 yield

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.

Reduction to Operator Equation
Let u R Ψ be a solution to 5.2 by u 0 0 and given Ψ ∈ L 2 Q T .Obviously, the expression u R Ψ defines a bounded linear operator from

6.1
Thus, the problem 1.1 -1.2 , 2.1 with u 0 0 is reduced to equation We show that for sufficiently large R and sufficiently small T > 0, operator equation 6.2 is uniquely solvable in 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 u ∈ S R .Properties of the operator R and 6.1 imply u 0, x 0 and Let us show that From 5.10 , it follows that Hence, from 5.6 , the Hölder inequality, and u 1,2 ≤ R, it follows that

7.4
Estimate S 2 0 .Differentiating and applying the integral Minkowski inequality, we have Making the change y z s; t, x , we get |u x s, z s; t, x | 0,t u x s, y 0,s .7.6 From 3.3 , it follows that From this by means of H ölder inequality, it follows that 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 W 1,2 2 Q T , it follows that the metric space S R is complete 10, Chapter 3, Section 1 .

ISRN Mathematical Physics
By means of simple calculuses, we obtain

7.11
It follows from the continuous imbedding W 1 2 Ω t ⊂ L 4 Ω t , 4.8 , and 5.6 that The inequality 4.2 for ψ |u i t, x | 2 and 4.3 by u i t, x 0 on S T give

7.13
Hence, from u i 0, x 0 and 5.6 , we obtain

7.15
Let us estimate RZ 2 .As by the evaluation of RZ 1 , we have

7.17
Let us estimate the first term.As in 10, Chapter 3, Section 1.2 , we continue the u 1 and u 2 out of Q T on R n with the preservation of class so that the velocity fields v 1 u 1 w and v 2 u 2 w of the corresponding Cauchy problems 2.1 are vanished out of a compact convex domain Ω, Ω ⊃ Ω t , 0 ≤ t ≤ T .By the uniqueness theorem for Cauchy problems 7.11 , solutions z 1 and z 2 are the restrictions of the corresponding extended Cauchy problems.We mark the extended u 1 , u 2 , z 1 , and z 2 by the bar at the top.
We now estimate G 1 .Using the Newton-Leibnitz formula, we have

7.18
Here, Y y s, t, x, α αz 1 s; t, x 1 − α z 2 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 0 ≤ α ≤ 1, 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

7.19
Using the change of variable y y s, t, x, α , we get

7.22
A similar estimate for G 2 is established easier.From the estimates 7.16 , 7.21 , 7.22 , the inequality A similar estimate for RZ 3 0,1 is established similarly.From the estimates RZ i 0,1 for i 1, 2, 3, it follows the inequality where q ∈ 0, 1 by sufficiently small T .Lemma 7.2 is proved.

7.28
It follows from this and 4.17 that by g, v ∈ W 1,2 2 Q T , the inequality holds.Setting g w, v u 0 , we obtain the required inclusion.Note that this and 7.25 imply the estimates

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 F 0 , w Q k , replacing by k > 1 the initial condition u 0, x u 0 x by the initial conditions u k t k , x u k−1 t k , x , x ∈ Ω k .From the local solvability, which is established above, it follows that the length of the segment T k , t k 1 , at which these problems are uniquely solvable are determined by F L 2 Q k and |U k−1 t k , x | 1,t k .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 t k , t k 1 .We solve these problems sequentially, starting with k 0. By a priori estimate 5.1 , |u k−1 t k , x | 1,t k is uniformly bounded on t.In addition, it is obvious that F L 2 Q k ≤ F L 2 Q T .Thus, we obtain the solvability to problems for all k.The function u t, x , whose restriction to t k , t k 1 is u k t, x , is obviously the unique solution to 2.3 -2.4 , 2.1 at Q T .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.

2 Q T , and the estimate u 1 , 2 ≤ M w 1 , 2 , 1 , 2 ≤ M w 1
.3 -2.4 has a unique solution u ∈ W 1,2 follows from 4 that the similar result for more simple linear problem 5.2 holds, by this in our case, the estimate u 14 that for functions u ∈ W1,2   2 •, • t the inner product in L 2 Ω t , H t and V t are the closure of smooth compactly supported solenoidal functions in Ω t in the norm of L 2 Ω t and W 1 2 Ω t , respectively.We denote by P t the orthogonal projector in L 2 Ω t on H t .Consider in H t unbounded linear operator A t v −P t Δv, with the domain D A t τ t v s, z s; t, x ds, τ ∈ 0, T , T, x ∈ Q T , 2.1 3 , we mean a continuous with respect to all variables vector function z τ; t, x , satisfying 1.3 .Denote by u x a Jacobi matrix of the vector function u x , by u xx a tensor which is composed of second derivatives of the components of a vector function u x , and by |u x | 2 0 and |u xx | 2 0 the sum of squares of L 2 Ω -norms of the first and second derivatives, respectively.The following result was established in 1 for a cylindrical domain Q T Multiply both sides of 2.3 in L 2 Ω t by u.The standard arguments as in 5 and 4.
t , 4.3 where Ψ t, x u t, x • u t, x , u ∈ W 1,2 2 Q T , and vanishes on S T .i t, x ∂u t, x /∂x i , w t, x t for v, u, w ∈ W 1 2 Ω t .Since div w 0, then B t w, u, u t .4.6 Using the inequality | u, v t | ≤ |u| −1,t |v| 1,t see 7, page 252 and the boundedness of the operator ∂/∂x i : W 1 2 Ω t → L 2 Ω t , we have |b t w, u, u | ≤ M ∂ w i u ∂x i −1,t |u| 1,t ≤ M|wu| 0,t |u| 1,t .4.7 With the help of H ölder's inequality and holds for b t 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 t w, u, u | ≤ M w L 4 Ω t |u| 3/2 1,t |u| 1/2 0,t ≤ ε|u| 2 1,t M 2 ε w 4 L 4 Ω t |u| 2 0,t .4.10A similar estimate 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 Q t : ≤ M|u s, x | 0,s |u t, x | 2 a,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 2 ds.
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 u 0 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 u 0 / 0. Let u 0 t, x be a solution to the linear problem 5.2 by u 0 / 0 and F 0. There exists only one solution to this problem in the force of Lemma 5.2, and the inequality Lemma 7.3.For sufficiently large R and sufficiently small T , the operator Q in S R has a unique fixed point.Proof.andz is a solution to problem 2.1 by v u w u 0 .In this case, F, w ∈ L 2 Q T .In fact, by g, v ∈ L 4 Q T with the help of H ölder's inequality, we have 1,2 , there exists a unique solution u to problem 7.26 , 2.1 , and the inequality 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 u 0 / 0 and the inequality supt |u t, x | 1,t u 1,2 ≤ M 16 F 1,2 , w 1,2 , u 0Now let T > 0 be arbitrary.Let T 0 such that our problem is uniquely solvable on 0, T 0 ⊂ 0, T .Note that the same result holds not only on 0, T 0 , but also on t * , t * T 0 ⊂ 0, T for anyt * > 0. Let T k kT /N,k 1, 2, . . ., N, N be a natural number.Consider problem 2.3 -2.4 , 2.1 on the domains Q k , where Q k { t, x : t k ≤ t ≤ t k 1 , x ∈ Ω t }.Let us find solutions u k t, x to problem 2.3 -2.4 , 2.1 from W 1,2 2 t |u t, x | 1,t u 1,2 ≤ M 15 F 1,2 , w 1,2 , u 0