Existence of Solutions for Hyperbolic System of Second Order Outside a Domain

Abstract

We study the mixed initial-boundary value problem for hyperbolic system of second order outside a closed domain. The existence of solutions to this problem is proved and the estimate for the regularity of solutions is given. The application of the existence theorem to elastrodynamics is discussed.

1. Introduction

This paper is concerned with the exterior problem for hyperbolic system of second order. Let be a closed domain with smooth boundary in and let the origin belong to . Consider the following exterior problem for the hyperbolic system of second order:

(1.1)

where and . We assume that satisfies

(1.2)

for all symmetric matrixes , where , .

Let . The system (1.1) can be written as an evolution system in the form

(1.3)

where

(1.4)

Ikawa considered in [1] the mixed problem of a hyperbolic equation of second-order. The existence theorem is known for the obstacle free problem in [2]. Dafermos and Hrusa proved in [3] the local existence of the Dirichlet problem for the hyperbolic system inside a domain by energy method.

In this paper, we deal with the exterior problem for the second order hyperbolic system. In Section 2, we show the existence of the exterior problem for the problem (1.1) by the semigroup theory. In Section 3, we prove the regularity for the solutions of the exterior problem (1.1) and give the estimate for the regularity of solutions. In Section 4, we discuss the application of the existence theorem to elastrodynamics.

2. Existence of the Exterior Problem for Hyperbolic System of Second Order

Note that with the inner product

(2.1)

By (1.2) and Korn inequality (cf. [4, 5]), we have

Lemma 2.1.

For some , we have

(2.2)

Then is a Hilbert space with the inner product defined as above. We define the operator (without loss of generality, we still write this operator as ) in by

(2.3)

where It is obvious that is a densely defined operator.

Lemma 2.2.

There exists a constant such that for any ,

(2.4)

holds.

Proof.

Let .

(2.5)

Corollary 2.3.

For all real such that , the estimate

(2.6)

holds for any .

Proof.

By (2.4),

(2.7)

The estimate of the resolvent operator is the following.

Lemma 2.4.

There exists a constant such that for all real and ,

(2.8)

is a bijective mapping. Moreover, we have

(2.9)

Proof.

Consider the system

(2.10)

namely,

(2.11)

where .

The substitution of the first relation

(2.12)

in the second of (2.11) gives

(2.13)

By the well-known variation method, there exists a solution of the elliptic system (2.13) for any . Defining by (2.12), we have a solution

(2.14)

of (2.10). Therefore, is a surjection.

From (2.6), it follows that the existence of and the estimate

(2.15)

Let , we have (2.9).

For , we define the following norm:

(2.16)

Suppose that , we have

Corollary 2.5.

For the real number ( fixed) and the integer , where is as in Lemma 2.4, there exists such that for any

(2.17)

Proof.

From Lemma 2.4,

(2.18)

is a bijective continuous mapping, then is a closed operator. It implies that is also a closed operator. By Banach's closed graph theorem, is continuous. So for any we have

(2.19)

Definition 2.6.

Let be a Banach space. A family of infinitesimal generators of semigroups on is called stable if there are constants and (called the stability constants) such that

(2.20)

for every finite sequence , .

Lemma 2.7.

For , let be the infinitesimal generators of semigroups on the . The family of generators is stable if and only if there are constants and such that

(2.21)

for any finite sequence ,

Lemma 2.8.

Let be a stable family of infinitesimal generators of semigroups on the space such that is independent of and for every , is continuously differentiable in . If , then

(2.22)

has a unique classical solution such that

The proofs of Lemmas 2.7 and 2.8 are in [6]. The straightforward application of the semigroup theory to the system (1.3) gives the following proposition.

Proposition 2.9.

Given and , then there exists one and only one solution of (1.3) such that .

Proof.

Let . For given , is an infinitesimal generator of semigroups on . For any , it is easy to know that

(2.23)

Then for any , we have

(2.24)

namely,

(2.25)

For any finite sequence and any , ,

(2.26)

where . From Lemma 2.4, for any , . Then by Lemma 2.7, is a stable family. Obviously, is continuously differentiable in . So Proposition 2.9 follows from Lemma 2.8.

From Proposition 2.9, we obtain the existence of solutions to the problem (1.1).

Theorem 2.10.

Given and , then there exists one and only one solution of (1.1) such that

(2.27)

Proof.

Let , . By Proposition 2.9, there exists a solution of problem (1.3) such that . Let denote the forgoing three components of , then is the solution of problem (1.1) and satisfies (2.27).

3. Regularity of Solutions for the Exterior Problem

First, we show the energy inequalities for our problem. These inequalities play an important role in the proof of the regularity of solutions.

Proposition 3.1.

Suppose that

(3.1)

is a solution of problem (1.1) and that , then for any given , we have

(3.2)

where is a constant which depends on .

Proof.

Put , then and satisfies

(3.3)

where , . Obviously,

(3.4)

By (2.4),

(3.5)

Thus

(3.6)

Applying Gronwall's inequality, we get

(3.7)

Without loss of generality, we assume that . Then we see

(3.8)

Applying (3.7) for , we get

(3.9)

By (2.17) and (2.2),

(3.10)

Obviously,

(3.11)

Also we have

(3.12)

and for all ,

(3.13)

Inserting these estimates to the above inequality, we get

(3.14)

An application of Gronwall's inequality implies

(3.15)

Namely,

(3.16)

This completes the proof of (3.2).

Theorem 3.2.

For , suppose that , , , and

(3.17)

If the compatibility conditions of order are satisfied, then problem (1.1) has a solution such that

(3.18)

Proof.

At first we prove

(3.19)

Let and . We define by

(3.20)

then ,

We consider the following problem:

(3.21)

where

(3.22)

here .

From (3.21),

(3.23)

By (3.2), we have

(3.24)

thus

(3.25)

This implies that converges to some in . Set

(3.26)

then tends to in . The passage to the limit of (3.21) shows

(3.27)

namely,

(3.28)

Taking account of the definition of , we see

(3.29)

Therefore is the solution of problem (1.1) and satisfies (3.19).

We now prove (3.18) by induction. When , (3.18) follows from (3.2). For , suppose that (3.18) holds for . We show that it still holds for .

Applying (3.2) to (3.27), we conclude from the inductive hypothesis that

(3.30)

In a similar way, we can obtain

(3.31)

Set , then is the solution of (1.3) and

(3.32)

Now

(3.33)

then by (2.17) (taking ), we see

(3.34)

Differentiation of (3.33) with respect to gives

(3.35)

and by the above result ,

(3.36)

from which it follows that

(3.37)

Repeating this process, we get

(3.38)

Using this, we see the right-hand side of (3.33) , then by (2.17) (taking )

(3.39)

This assures that the right-hand side of (3.35) , then

(3.40)

Repeating this process, we get

(3.41)

Step by step, finally, we get

(3.42)

and (3.18).

4. Application to Elastrodynamics

It is well known that the displacement of an isotropic, homogeneous, hyperelastic material without the action of external force satisfies the following hyperbolic system (cf. [4, 5]):

(4.1)

where , and , are given by the Lamé constants , :

(4.2)

We assume that ,

From [5], system (4.1) can be written as

(4.3)

where stands for the elastic tensor.

The system (4.3) is the special case of the system (1.1). So by the existence Theorem 3.2, we derive the existence of solutions for the initial-boundary problem to the elastrodynamic system (4.3) outside a domain.