On Global Existence of Solutions of the Neumann Problem for Spherically Symmetric Nonlinear Viscoelasticity in a Ball
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.
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. ) and Kawashima and Shibata (cf. ) for quasilinear hyperbolic system of second-order with viscosity.
In the paper of , 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 , Pawłow and Zajączkowski showed the existence and uniqueness of global regular solutions to the Cahn-Hilliard system coupled with viscoelasticity.
Finally, in , 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 . 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
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 . 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  for and in  for any .
The weighted Sobolev spaces with fractional derivatives are introduced in .
In the nonstationary case, Lemma 1 follows from the methods described in  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
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 ,.
Finally , , and so on.
To prove the Main Theorem we have to recall some estimates proved in .
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
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
To calculate the above expression we need problems for and . From Lemma 3.4 from , 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.
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).
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
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
From the expression of , we have
In view of (82) and (83), we obtain
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
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
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.
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.
Let (105) hold. Let
Then solutions to problems (11)–(13) satisfy where , is from (36), from (40) and
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