ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 268505, 13 pages

http://dx.doi.org/10.1155/2013/268505

Research Article

On Global Existence of Solutions of the Neumann Problem for Spherically Symmetric Nonlinear Viscoelasticity in a Ball

Jerzy A. Gawinecki¹ and Wojciech M. Zajączkowski1,2

¹Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, Kaliskiego 2, 00-908 Warsaw, Poland

²Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland

Received 21 November 2012; Accepted 20 December 2012

Academic Editors: G. Akrivis, L. Wang, and C. Zhu

Copyright © 2013 Jerzy A. Gawinecki and Wojciech M. Zajączkowski. 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 examine spherically symmetric solutions to the viscoelasticity system in a ball with the Neumann boundary conditions. Imposing some growth restrictions on the nonlinear part of the stress tensor, we prove the existence of global regular solutions for large data in the weighted Sobolev spaces, where the weight is a power function of the distance to the centre of the ball. First, we prove a global a priori estimate. Then existence is proved by the method of successive approximations and appropriate time extension.

1. Introduction

First, we recall some important facts from the nonlinear theory of viscoelasticity. Among the papers devoted to nonlinear viscoelasticity, we mention below some of them. The global solution (in time) for sufficiently small and smooth data are proved by Ponce (cf. [1]) and Kawashima and Shibata (cf. [2]) for quasilinear hyperbolic system of second-order with viscosity.

In the paper of [3], Kobayashi, Pecher, and Shibata proved global in time solution to a nonlinear wave equation with viscoelasticity under the special assumption about nonlinearity. In the paper of [4], Pawłow and Zajączkowski showed the existence and uniqueness of global regular solutions to the Cahn-Hilliard system coupled with viscoelasticity.

Finally, in [5], global existence of regular solutions to one-dimensional viscoelasticity is proved. Moreover, many facts on elasticity and viscoelasticity theory can be found in [6–10]. Some existence results in the linear and nonlinear thermoviscoelasticity are shown in [11, 12].

In our paper, we consider a more general nonlinear system of viscoelasticity because the stress tensor is a general nonlinear function depending on a strain. We assume that the stress tensor is a function of a strain at a given instant of time , but it does not depend on strains at time . It is worth to emphasize that our constitutive relation for the stress tensor and any another constitutive relation satisfy the rules of continuum mechanics.

In order to prove the global (in time) solution for nonsmall data for nonlinear system of viscoelasticity (cf. formulae (1), (2), and (3)), we consider the spherically symmetric case and use the anisotropic Sobolev spaces with weights.

Speaking more precisely, we consider the motion of viscoelastic medium described by the following system of equations (cf. [7, 11]):

where is the displacement vector, is a given system of the Cartesian coordinates, , is the mass density, is the stress tensor, and is the external force field.

We examine system (1) in a bounded domain with the Neumann boundary conditions

where , is the unit outward vector normal to .

Moreover, we add the initial conditions

We will assume that

where is the linearized strain tensor, is some function which will be specified later, and is a positive constant.

Our aim is to prove the global existence of solutions to problems (1)–(4) for nonsmall data.

Since we do not know how to show the existence in a general case, we restrict our considerations to the spherically symmetric case. We assume that is a ball with radius centered at the origin of the introduced Cartesian coordinates. We introduce the spherical coordinates by the relations

With these coordinates, we connect the orthonormal vectors

Then, we define , , , , , . Since the spherically symmetric case is considered, we have .

To simplify the notation, we introduce

Assuming and transforming (1) to the spherical coordinates, we obtain

where

Let us introduce the quantity

where .

Then, (8) takes the form

and in view of (3), we have the initial conditions

and in view of (2) and (9), the boundary condition

where .

Global existence of regular solutions to problems (11) and (12) with the Dirichlet boundary condition

is proved in [13]. To formulate the main results of this paper we need the following.

Assumptions. Let us introduce the notation , . Assume that

(1), (2)there exist positive constants , , , such that

(3),, ,(4),, , , , , ,(5), , .

Main Theorem. (1) Let Assumptions – hold. Then there exists a solution to problems (11)–(13) such that

(2) Let Assumptions (1) and (4) hold, then

(3) Let Assumptions (1), (4), and (5) be satisfied. Then

Our paper is organized as follows. In Section 1, the formulation of the considered problem and the main results are presented. In Section 2, the notation is introduced. Mainly, we define the anisotropic Sobolev spaces with weights. Section 3 is devoted to the proof of energy-type estimates to solutions of problems (11)–(13).

In Section 4 the existence of the global solution for nonsmall data of problems (11)–(13) is proved.

2. Notation and Auxiliary Results

By we denote the generic constant which changes from formula to formula. By , , we denote a generic function which is always positive and increasing.

We replace forms of right-hand side (left-hand side) by the abbreviation r.h.s. (l.h.s.). We mark , , and so on.

By we denote the space of bounded functions on the interval .

By , , , we denote a weighted Sobolev space with the finite norm

and .

By , we denote the Hölder space with the finite norm

Next, we recall the Hardy inequality (see [14, Chapter 1, Section  2.15])

where , , and.

The inequality holds also for functions with compact support. Assuming that , we introduce and repeat the proof from [14, Chapter 1, Section  2.15].

From [15, Chapter 2, Section  3], we have the imbedding

where .

Finally, we consider the problem

To examine nonstationary problem (23), we need anisotropic weighted Sobolev spaces , , , of functions with the finite norm

Spaces appropriate for elliptic problems were introduced in [16]. Moreover, we assume that .

The following result is valid.

Lemma 1. We assume that , . Then there exists a solution to problem (23) such that and

In the case of elliptic equations, a similar result was proved in [17] for and in [16] for any .

The weighted Sobolev spaces with fractional derivatives are introduced in [16].

In the nonstationary case, Lemma 1 follows from the methods described in [18] in the case . For the general Lemma 1 results from considerations in [18–20].

Finally, we introduce spaces used in this paper. We will define them by introducing finite norms.

Besov space , , , ,

where is the integer part of ,

where and

for integer.

is the space of bounded functions.

, , , are the Hölder spaces with the finite norms

where is the closure of .

By , , , we denote a space of functions with the finite norm

We introduce also the Sobolev spaces

where , ,

where , , , ,

, . For , we have , , , is the Besov space introduced in [14, Chapter 4, Section  18] and is the Sobolev-Slobodetsky space, where ,.

Moreover,

Finally , , and so on.

3. Estimates

To prove the Main Theorem we have to recall some estimates proved in [13].

Lemma 2 (see [13, Lemma  3.1]). We consider problems (11)–(13). Assume that , are initial data defined by (12). Let be a constant such that

where is a positive function differentiable with respect to its arguments introduced by (10).

Then, solutions to problems (11)–(13) satisfy the estimate

Next, we need the following.

Lemma 3 (see [13, Lemma  3.2]). We consider problems (11)–(13). Let , . Assume that . Assume existence of positive constants , , , and such that

Finally, assume that

Then solutions to problems (11)–(13) satisfy the following estimate:

where , .

Continuing, we have the following.

Lemma 4 (see [13, Lemma  3.3]). We consider problems (11)–(13). Assume that there exist positive constants , , , , and such that

Assume also that

where

Then solutions to (11)–(13) satisfy the following inequality:

where , and

Remark 5. In general constants and , , are different. Let us consider the example

where is a positive constant. Then we have

Continuing, we derive

For we obtain . But for we have

It is clear that many examples can be invented.

We need to obtain an estimate from (44). For this purpose, we examine the last two norms on the r.h.s. of (44). We express them in the form

Hence, by the Hardy inequality (see [14, Chapter 1, Section  2.15]), the above expression is bounded by

To estimate , we need the Pego transformation

Hence,

To calculate the above expression we need problems for and . From Lemma 3.4 from [13], we have the following problems for and :

Lemma 6. Let the assumptions of Lemma 3 be satisfied. Let , , means the fractional derivative. Then solutions to problems (54) and (55) satisfy

where is introduced in (40), , , , , and is introduced in Lemma 3.

Proof. For solutions to problem (54), we have (see Lemma 1),

Now, we examine the terms from the r.h.s. of (58).

The first norm on the r.h.s. of (58) equals

Applying the Hölder inequality, the second integral on the r.h.s. of (58) is bounded by

where the last inequality holds in virtue of Lemma 3 and under the assumption

where is introduced in Lemma 3. Hence,

The third integral on the r.h.s. of (58) we express in the form

Assuming , setting , and recalling imbedding (22) and Lemma 3, we obtain for the estimate

Finally, the last term on the r.h.s. of (58) is bounded by

Using the above estimates in the r.h.s. of (58) yields (56).

Repeating the considerations leading to (56) to problem (55) gives (57). This concludes the proof.

Using (56) and (57) in (53) gives

Applying the equality

in the first term on the r.h.s. of (66) and using the Gronwall lemma, we get

Now, we have to estimate the norm on the r.h.s. of (68). For the purpose we need the following.

Lemma 7. Let be a solution to (11)–(13). Assume that and (37), (38) are satisfied. Assume that , , where , . Then

where is introduced in (36).

Proof. Let us introduce a smooth function such that for and for , where .

First, we obtain a local estimate in for solutions to problems (11)–(13). Multiplying (11) by and integrating over yields

Introducing the notation

we obtain

Continuing, we have

Applying the Hölder and the Young inequalities to the r.h.s. of the above equality and using (38) yields

Using that in the above inequality implies

Integrating the above inequality with respect to time and using (37), we have

Since , the above inequality yields

where (36) was used again.

Now, we introduce the following Pego transformation

Moreover, we have the initial-boundary value problem for ,

The nonhomogeneous Dirichlet boundary condition is not convenient so we introduce the new function

which is a solution to the problem

We have to calculate . For this purpose, we need the expressions which follow from (11) multiplied by

and from (13) multiplied by

Using (82) and (83) in yields

For solutions to problem (81), we have

Hence,

From the expression of , we have

In view of (82) and (83), we obtain

Hence,

Employing (38) yields

From the form of (see (81)₂), we derive

Using (90) in (86) yields

Since and that

we obtain from (92) the inequality

where the relation was exploited.

Using the interpolation inequality

in (94), assuming that is sufficiently small, and using (77), we obtain

Employing

in (96) and applying the Gronwall lemma, we get

From (98), we obtain (69). This concludes the proof.

To derive an estimate for (44), we have to take into account Lemmas 6 and 7. The lemmas imply an estimate for the first two terms on the r.h.s. of (44). For this purpose, we need the following.

Lemma 8. Let be a solution to problems (11)–(13). Assume that there exist constants , , , such that (41) holds, where , , . Assume that , , , is defined in (36),

Then the following estimate holds:

Proof. From (44) and (51), we have

where

Inequality (68) takes the form

where , .

Finally, (69) yields

To apply (102) and (103) in the r.h.s. of (101), we assume , . In view of the assumptions of Lemma 6, we have that so . This implies that . Hence, . Since , assumptions of Lemma 6 imply that , so . Employing (102) and (103) in (101) yields (100). This concludes the proof.

Remark 9. In view of the expression of appeared in the assumptions of Lemma 4 we derive

Since can be chosen positive, we see that all weights in the norms , can be chosen as power functions with positive exponents.

Assuming that the r.h.s. of (104) is finite assumptions (99) can be replaced by

Up to now there is no estimate for . Hence, to get the space regularity, we need the following.

Lemma 10. Let (105) hold. Let

Then solutions to problems (11)–(13) satisfy

where , is from (36), from (40) and

Proof. Multiplying (11) by and integrating over , we obtain

Continuing, we have

Using (106), we obtain

Integrating the inequality with respect to time, assuming that is sufficiently small, using that , , and employing (40) with and (102), (103) with , , , we obtain (101). Hence, . This concludes the proof.

Finally, we recall some local properties of solutions to problems (11)–(13) which are proved in Lemma 3.11 in [13].

Lemma 11. Let the assumptions of Lemma 3 be satisfied. Then

4. Existence

We prove the existence of solutions to problems (11)–(13) by the following method of successive approximations:

where , and we assume that the zero approximation is defined by

The boundary condition (113)₃ implies (13) because

First, we examine the linear problem

Lemma 12. Assume that , , . Then there exists a solution to problem (116) such that and the estimate holds

Proof. The existence can be proved by the Galerkin method. Therefore, we shall restrict the proof to show only estimate (117).

Multiplying (116)₁ by and integrating the result over yields

Looking for solutions that is finite at , the last but one term on the l.h.s. of (118) vanishes for . Vanishing of the term can be guaranteed by choosing an appropriate base functions.

Integrating by parts in the third term on the l.h.s. of (118) implies

For , the last term on the l.h.s. is positive so it can be omitted in the estimate implied by (119). Then we show the existence of solutions such that

Hence, near , where . Then the last term on the l.h.s. of (119) vanishes. For , the last term on the l.h.s. of (119) vanishes under the assumption that we are looking for solutions such that near , .

Finally, we estimate the first term on the r.h.s. of (119). Let us recall the set

Then by imbedding, we get

where .

Using that is sufficiently small, we obtain

Integrating (123) with respect to time, we get the estimate

The above inequality implies the estimate for . To increase the interval, we use the Hardy inequality

Combining (124) and (125), we see that (124) gives an estimate for satisfying the inequality

Solving (126), we see that . Since is very close to 1, we can assume that

Hence, .

Finally, for satisfying (127), we have estimate (124).

In the next step, we consider the parabolic problem

For solutions to problem (128), we have the inequality (see Lemma 1)

The first two norms on the r.h.s. of (129) can be estimated in view of (124). Then we obtain some restriction on . We have

where the last inequality holds for, so . Hence, can be chosen less than 1. Similarly, the inequalities

hold for so . Hence, the same restriction on as before is satisfied. Since the existence follows from the Galerkin method, Lemma 12 is proved.

In view of Lemma 12, we have the following.

Lemma 13. Assume that , ,. Assume the Lipschitz continuity

Then there exists a solution to problems (11)–(13) such that and

Proof. Applying Lemma 12 to problem (113) yields

where and we used that

In view of the relation

we obtain from (134) the inequality

Hence, for sufficiently small , we have

for any .

To show convergence of the sequence we introduce the differences

which are solutions to the problems

To show convergence we multiply (140) by and integrate the result over . Then we have

Integrating by parts in yields

where

and the second integral in view of the assumption of the Lipschitz continuity is bounded by

Next, we examine

Finally, we estimate ,

Using the above estimates in (141) and assuming that are sufficiently small, we obtain

where and the same considerations as in (125)–(127) are repeated. Integrating (147) with respect to time and employing the formula

we derive from (147) the inequality

Hence, we have that the sequence converges and then we show the existence of solutions to problems (11)–(13) on a small interval . Since we have the estimate on the interval with arbitrary , the above considerations can be repeated step by step.

In view of Lemmas 8 and 10 and Remark 9, the extension can be made under the restriction that . This concludes the proof.

Proof. Proof of the Main Theorem. In view of assumptions of Lemmas 8 and 10, Remark 9, and also by Lemmas 12 and 13, we have the first assertion of the theorem. In view of Lemma 3 and the Galerkin method, we prove the second assertion. In view of Lemmas 3 and 11, the third assertion follows. This concludes the proof.

Acknowledgment

W. M. Zajączkowski is partially supported by MNiSW Grant no. NN 201 396 937.

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