Pointwise Estimates for Solutions to a System of Radiating Gas

In this paper we focus on the initial value problem of a hyperbolic-elliptic coupled system in multi-dimensional space of a radiating gas. By using the method of Green function combined with Fourier analysis, we obtain the pointwise decay estimates of solutions to the problem.


Introduction
In this paper we consider the initial value problem , here is a constant vector, and are unknown functions of Typically, represent the velocity and radiating heat flux of the gas respectively., u q The system (1.1) is a simplified version of the model for the motion of radiating gas in n-dimensional space.More precisely, in a certain physical situation, the system (1.1) gives a good approximation to the following system describing the motion of radiating gas, which is a quite general model for compressible gas dynamics where heat radiative transfer phenomena are taken into account, where ρ, u, p, e and θ are respectively the mass density, velocity, pressure, internal energy and absolute temperature of the gas, while q is the radiative heat flux, and a 1 and a 2 are given positive constants depending on the gas itself.The first three equations are motivated by the usual Euler system, which describe the in-viscid flow of a compressible fluid and express conservation of mass, momentum and energy respectively.We refer to the book of Courant and Friedrichs [1] for a detailed derivation of several models in compressible gas-dynamics.The physical motivation of the fourth equation, which takes into account of heat radiation phenomena, is given in [2].
Concerning the investigation on the hyperbolic-elliptic coupled system in one-dimensional radiating gas, we refer to [5,6].In the case of the muti-dimensional case, Francesco in [7] obtained the global well-posedness of the system (1.1) and analyzed the relaxation limits.Recently, in [8], Liu and Kawashima investigated the decay rate to diffusion wave for the initial value problem (1.1) in n(n ≥ 1)-dimensional space by using a time-weighted energy method.
The rest of the paper is arranged as follows.Section 2 gives the full statement of our main theorem.In Section 3, we give estimates on the Green function by Fourier analysis which will be used in Section 5. Section 4 gives the global existence of solutions to the problem (2.3).In Section 5, we obtain the pointwise decay estimates of solutions.
Before closing this section, we give some notations to be used below.Let


and we denote its inverse transform as 1 .
is the usual Lebesgue space with the norm .

Main Theorems and Proof
For simplicity, without loss of generality, we . That is, we will consider the following initial value problem: Our Main results are the following: be an integer.
Assume that and put Then there is a small positive constant such that if 0 0 then the problem (2.3) has a unique global solution with 0, ; , 0, ; , 0, ; 0, ; .
and for any multi-indexes , Remark.In Theorem 2.1, we do not need to assume that The results in Corollary 2.2 is similar to those in [7].

The Global Existence of Solution
This section is devoted to prove the global existence result stated in Theorem 2.1.In [7], the global existence of solutions to the problem (2.3) is obtained, but for the completeness of this paper, here we give the sketch of the proof.
Since a local existence result can be obtained by the standard method based on the successive approximation sequence, we omit its details and only derive the desired a priori estimates of solutions.
Now we make energy estimates by using (3.4) under the following a priori estimate:   is a given constant.
Multiplying (3.4) by and integrating with respect to u x , by integration by parts we have that  by and integrating with respect to x , by integration by parts we have that We add up (3.7) with 1 l s    and get that In view of (3.5), (3.9) yields that By the continuity argument, we have the following result.

Estimates on Green Function
In order to study the problem (2.3), we start with the Green function (or the fundamental solution) to the linear problem corresponding to the Equation (3.4), which satisfies .
By Fourier transform we get that, By direct calculation we have that   the smooth cut-off functions, where  and are any fixed positive numbers satisfying We are going to study  First we give a lemma which is important for us to make estimates on the low frequency part.
If has compact support in the variables  , is a positive integer, and there exists a constant such that where and are any fixed integers, , , Proof.For  being sufficiently small, by noticing that is a smooth function of using Taylor expansion we have that Thus we complete the proof of Proposition 4.2. As for we have the following estimates.
   Assume that, if which is true as 0   by (4.14)By using (4.13), we have the following problem for By multiplying (4.17), whose variables are now and integrating over the region , we have that In view of (4.16) for 1, l


, then there exist distributions where   x  is the Dirac function.Furthermore, for positive integer 2  , with 0  being sufficiently small.
The proof of Lemma 4.4 can be seen in [9].
Choose sufficiently large such that R It is obvious that by direct calculation, we have that for 1, By using Lemma 4.4 we have the following result.Proposition 4.5.For being sufficiently large, there exist distributions and constant such that where   is an arbitrary positive integer.

Pointwise Estimates
In this section, we focus on the pointwise estimates of solutions to the problem (2.3).By Duhamel principle, the solution to the Equation (3.4) with initial datum can be expressed as following, Now we give a lemma which will be used in the following analysis.
Lemma 5.1.When 1 2 , n n n  2, and The proof of Lemma 5.1 can be seen in [9].
  , by using Lemma 5.1 and Theorem 4.6 with , we have that Thus we obtain that

Next we come to make estimates on
To this end, we will use the following lemma.

 
, .u x t  Lemma 5.2.Assume , then the following inequalities hold, , and 2  A t  , then Now we come to make estimates to by using   , u x t  Theorem 4.6 with and Lemma 5.2.We decompose


Next we estimate respectively by using Theorem 4.6.By using Lemma 5.2 (1), we have that x t  .By using Lemma 5.2 (2), we have that

I CM T t B x y t B y y CM T t B y CM T B x t t
Combining the two cases, we have that      Combining the two cases, we have that As for 42 , I by direct calculation, we have that To estimate 5 , I we will use the following result, which is obtained in [7]., , 0, 3.
Without loss of generality, we assume that then Combined with Proposition 4.5 and the fact that

Proposition 4 . 2 .
we have that Thus we complete the proof. By using Lemma 4.1 we can get the following proposition about the estimates on For sufficiently small  ,

1 .
In view of(5.19) and (5.20), we get that dependence on the initial data.In view of Theorem 3.1, the proof of Theorem 2.1 is completed.


Thus we complete the proof of Proposition 4.3. .Proof.For any fixed  , we choose sufficiently m .