We show that the energy to the thermoelastic transmission problem decays exponentially as time goes to infinity. We also prove the existence, uniqueness, and regularity of the solution to the system.
1. Introduction
In this paper we deal with the theory of thermoelasticity. We consider the following transmission problem between two thermoelastic materials:
We denote by a point of () while stands for the time variable. The displacement in the thermoelasticity parts is denoted by , (, ) and , (, ), , and is the variation of temperature between the actual state and a reference temperature, respectively. , are the thermal conductivity. All the constants of the system are positive. Let us consider an dimensional body which is configured in ().
The thermoelastic parts are given by and , respectively. The constants are the coupling parameters depending on the material properties. The boundary of is denoted by and the boundary of by . We will consider the boundaries and of class in the rest of this paper. The thermoelastic parts are given by and , respectively, that is (see Figure 1),
We consider for the operators
where , () are the Lamé moduli satisfying .
Figure 1. Domains and and boundaries of the transmission problem.
The initial conditions are given by
The system is subject to the following boundary conditions:
and transmission conditions
The transmission conditions are imposed, that express the continuity of the medium and the equilibrium of the forces acting on it. The discontinuity of the coefficients of the equations corresponds to the fact that the medium consists of two physically different materials.
Since the domain is composed of two different materials, its density is not necessarily a continuous function, and since the stressstrain relation changes from the thermoelastic parts, the corresponding model is not continuous. Taking in consideration this, the mathematical problem that deals with this type of situation is called a transmission problem. From a mathematical point of view, the transmission problem is described by a system of partial differential equations with discontinuous coefficients. The model (1.1)–(1.13) to consider is interesting because we deal with composite materials. From the economical and the strategic point of view, materials are mixed with others in order to get another more convenient material for industry (see [1–3] and references therein). Our purpose in this work is to investigate that the solution of the symmetrical transmission problem decays exponentially as time tends to infinity, no matter how small is the size of the thermoelastic parts. The transmission problem has been of interest to many authors, for instance, in the onedimensional thermoelastic composite case, we can refer to the papers [4–7]. In the two, three or dimensional, we refer the reader to the papers [8, 9] and references therein. The method used here is based on energy estimates applied to nonlinear problems, and the differential inequality is obtained by exploiting the symmetry of the solutions and applying techniques for the elastic wave equations, which solve the exponential stability produced by the boundary terms in the interface of the material. This methods allow us to find a Lyapunov functional equivalent to the secondorder energy for which we have that
In spite of the obvious importance of the subject in applications, there are relatively few mathematical results about general transmission problem for composite materials. For this reason we study this topic here.
This paper is organized as follows. Before describing the main results, in Section 2, we briefly outline the notation and terminology to be used later on and we present some lemmas. In Section 3 we prove the existence and regularity of radially symmetric solutions to the transmission problem. In Section 4 we show the exponential decay of the solutions and we prove the main theorem.
2. Preliminaries
We will use the following standard notation. Let be a domain in . For , are all real valued measurable functions on such that is integrable for and is finite for . The norm will be written as
For a nonnegative integer and , we denote by the Sobolev space of functions in having all derivatives of order belonging to . The norm in is given by . with norm , with norm . We write for the space of valued functions which are times continuously differentiable (resp. square integrable) in , where is an interval, is a Banach space, and is a nonnegative integer. We denote by the set of orthogonal real matrices and by the set of matrices in which have determinant 1.
The following results are going to be used several times from now on. The proof can be found in [10].
Lemma 2.1.
Let for or for be arbitrary but fixed. Assume that , , , , , and satisfy
Then the solution , , , and of (1.1)–(1.13) has the form
where , for some functions , , , and .
Lemma 2.2.
One supposes that is a radially symmetric function satisfying . Then there exists a positive constant such that
Moreover one has the following estimate at the boundary:
Remark 2.3.
From (2.3) we have that
The following straightforward calculations are going to be used several times from now on.
(a) From (1.8) we obtain
(b) Using (1.10) and (1.11) we have that
(c) Using (1.6) we have that
Thus, using (1.10) and (1.11) we have that
Similarly, we obtain
Throughout this paper is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities.
3. Existence and Uniqueness
In this section we establish the existence and uniqueness of solutions to the system (1.1)–(1.13). The proof is based using the standard Galerkin approximation and the elliptic regularity for transmission problem given in [11]. First of all, we define what we will understand for weak solution of the problem (1.1)–(1.13).
We introduce the following spaces:
for and .
Definition 3.1.
One says that is a weak solution of (1.1)–(1.13) if
satisfying the identities
for all , , , and almost every such that
The existence of solutions to the system (1.1)–(1.13) is given in the following theorem.
Theorem 3.2.
One considers the following initial data satisfying
Then there exists only one solution of the system (1.1)–(1.13) satisfying
Moreover, if
verifying the boundary conditions
and the transmission conditions
then the solution satisfies
Proof.
The existence of solutions follows using the standard Galerking approximation.
FaedoGalerkin Scheme
Given , denote by and the projections on the subspaces
of and , respectively. Let us write
where and satisfy
with
for almost all , where , , , and are the zero vectors in the respective spaces. Recasting exactly the classical FaedoGalerkin scheme, we get a system of ordinary differential equations in the variables and . According to the standard existence theory for ordinary differential equations there exists a continuous solution of this system, on some interval . The a priori estimates that follow imply that in fact .
Energy Estimates
Multiplying (3.14) by , summing up over , and integrating over we obtain
where
Multiplying (3.15) by , summing up over , and integrating over we obtain
where
Adding (3.17) with (3.19) we obtain
where
Integrating over , , we have that
Thus,
Hence,
In particular,
and it follows that
The system (1.1)–(1.4) is a linear system, and hence the rest of the proof of the existence of weak solution is a standard matter.
The uniqueness follows using the elliptic regularity for the elliptic transmission problem (see [11]).We suppose that there exist two solutions , , and we denote
Taking
we can see that satisfies (1.1)–(1.4). Since , are weak solutions of the system we have that satisfies
Using the elliptic regularity for the elliptic transmission problem we conclude that
Thus satisfies (1.1)–(1.4) in the strong sense. Multiplying (1.1) by , (1.2) by , (1.3) by , and (1.4) by and performing similar calculations as above we obtain , where
which implies that , , , and . The uniqueness follows.
To obtain more regularity, we differentiate the approximate system (1.1)–(1.4); then multiplying the resulting system by and and performing similar calculations as in (3.23) we have that
where
Therefore, we find that
Finally, our conclusion will follow by using the regularity result for the elliptic transmission problem (see [11]).
Remark 3.3.
To obtain higher regularity we introduce the following definition.
Definition 3.4.
One will say that the initial data is regular () if
where the values of and are given by
verifying the boundary conditions
and the transmission conditions
for . Using the above notation we say that if the initial data is regular, then we have that the solution satisfies
Using the same arguments as in Theorem 3.2, the result follows.
4. Exponential Stability
In this section we prove the exponential stability. The great difficulty here is to deal with the boundary terms in the interface of the material. This difficulty is solved using an observability result of the elastic wave equations together with the fact that the solution is radially symmetric.
Lemma 4.1.
Let one suppose that the initial data is 3regular; then the corresponding solution of the system (1.1)–(1.13) satisfies
where with
and .
Proof.
Multiplying (1.1) by , integrating in , and using (2.16) we have that
Multiplying (1.2) by , integrating in , and using (2.17) we have that
Multiplying (1.3) by , integrating in , and using (2.11) we have that
Multiplying (1.4) by , integrating in , using (2.11), and performing similar calculations as above we have that
Adding up (4.4), (4.5), (4.6), and (4.7) and using (1.12) and (1.13) we obtain
where
Thus
In a similar way we obtain (4.2).
Lemma 4.2.
Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.11) satisfies
where , are positive constants and
Proof.
Multiplying (1.1) by , integrating in , and using (1.10) we have that
Then
Hence
Thus
Hence
Therefore
Similarly, multiplying (1.2) by , integrating in , and performing similar calculations as above we obtain
Multiplying (1.3) by and integrating in we have that
Hence
Then
Using (1.10) and (2.9) and performing similar calculations as above we obtain
Replacing (1.1) in the above equation we obtain
On the other hand
Therefore
Multiplying (1.4) by , integrating in , and performing similar calculations as above we obtain
Adding (4.19) with (4.27) we have that
Adding (4.20) with (4.28) we have that
Moreover, by Lemma 2.2, there exist positive constants , such that
Therefore we obtain
Similarly
The result follows.
Lemma 4.3.
Under the same hypotheses of Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies
with
where , and are positive constants.
Proof.
Multiplying (1.1) by , integrating in , using (2.9), and performing straightforward calculations we have that
Using (1.10) we obtain
Multiplying (1.2) by , integrating in , and performing similar calculations as above we obtain
Multiplying (1.3) by and integrating in , we have that
Performing similar calculations as above we obtain
Multiplying (1.4) by , integrating in , and performing similar calculation as above we obtain
Adding (4.37), (4.38), (4.40), and (4.41), using (1.13), and performing straightforward calculations we obtain
with
Using the Cauchy inequality we have that
and, from trace and interpolation inequalities, we obtain
Similarly
Replacing in the above equation we obtain
The result follows.
We introduce the following integrals:
where
with , where is a ball with center and radius .
Lemma 4.4.
Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies
where , , and are positive constants and , .
Proof.
Using Lemma A.1, taking as above, , , and , we obtain
Applying the hypothesis on and since
we have that
Using (2.8) and the CauchySchwartz inequality in the last term and performing straightforward calculations we obtain
Finally, considering (1.1) and applying the trace theorem we obtain
with ; there exists a positive constant which proves (4.51).
We now introduce the integrals
Lemma 4.5.
With the same hypotheses as in Lemma 4.1, the following equality holds:
Proof.
Differentiating (1.2) in the variable we have that
Multiplying the above equation by and integrating in we obtain
Hence
On the other hand, using Lemma A.1 for , , , and we obtain
Multiplying (4.61) by and adding with (4.62) we obtain
The result follows.
We introduce the integral
where and are positive constants.
Lemma 4.6.
Under the same hypotheses as in Lemma 4.1 one has that the corresponding solution of the system (1.1)–(1.13) satisfies
Proof.
From (4.11), (4.12), and (4.34), using the CauchySchwartz inequality and performing straightforward calculations we have that
where , , and are positive constants. By Lemma 2.2, there exist positive constants and such that
Then
Hence, taking , , and we obtain
where we have used
Using (1.10), we have that
Thus
where .
We define the functional
where and are positive constants.
Theorem 4.7.
Let us suppose that is a strong solution of the system (1.1)–(1.13). Then there exist positive constants and such that
Proof.
We will assume that the initial data is 3regular. The conclusion will follow by standard density arguments. Using Lemmas 4.3 and 4.5 and considering boundary conditions, we find that
From (4.1), (4.2), and (4.75) we have that
where
Using the Cauchy inequality, we see that there exist positive constants , such that
Then . Note that for large enough we have that
From the above two inequalities our conclusion follows.
Acknowledgments
This work was done while the third author was visiting the Federal University of Viçosa. Viçosa, MG, Brazil and the National Laboratory for Scientific Computation (LNCC/MCT). This research was partially supported by PROSUL Project. Additionally, it has been supported by Fondecyt project no. 1110540, FONDAP and BASAL projects CMM, Universidad de Chile, and CI2MA, Universidad de Concepción.
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Appendix
We introduce the following functional:
where is a symmetric set of .
Lemma A.1.
Let be a radially symmetric set of . Suppose that and . Then for any function satisfying
where and are positive constants, one has that
where .
Proof.
We consider
Moreover
Hence
On the other hand,
Using
we obtain
Replacing in (A.6) the result follows.