Asymptotic Behavior of Solutions to a Model System of a Radiating Gas

In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates of solutions. Moreover, we show that the solution is asymptotic to the linear diffusion wave which is given in terms of the heat kernel.


Introduction
In this paper we consider the initial value problem: x ∈ R n , t > 0, (1.1) with the initial data u(x, 0) = u 0 (x), x ∈ R n . (1. 2) Here f j (u), j = 1, · · · , n, are smooth functions of u satisfying f j (u) = O(u 2 ) for u → 0, and u = u(x, t) and q = (q 1 , · · · , q n )(x, t) are unknown functions of x = (x 1 , · · · , x n ) ∈ R n (n ≥ 1) and t > 0. Typically, u and q represent the velocity and radiating heat flux of the gas, respectively. The system (1.1) is a simplified version of a radiating gas model in ndimensional space. More precisely, in a certain physical situation, the system (1.1) gives a good approximation to the following system of a radiating gas, that is a quite general model for compressible gas dynamics where the heat radiative transfer phenomena are taken into account: {ρ(e + |u| 2 2 )} t + div{ρu(e + |u| 2 2 ) + pu + q} = 0, −∇divq + a 1 q + a 2 ∇θ 4 = 0, (1.3) 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 form the usual Euler system, which describes the inviscid flow of a compressible fluid, and express the 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. On the other hand, the physical motivation of the fourth equation, which takes into account the heat radiation phenomena, is given in [32,3]. The simplified model (1.1) was first recovered by Hamer [8], and the reduction of the full system (1.3) to (1.1) was given in [32,8,16,7].
There are many works on the study of the hyperbolic-elliptic coupled system for one-dimensional radiating gas. The earlier paper with application to this kind of systems is [31], where Schochet and Tadmor studied the regularized Chapman-Enskog expansion for scalar conservation laws. We refer to [8,12,13,14,18,19,23,24,22,25,27,26] for shock waves, [5,10,9,11] for nonlinear diffusion waves, [16] for rarefaction waves, [15,17,21] for a singular limit and relaxation limit, and [4,29,30] for L 1 stability results, . In the multi-dimensional case, Di Francesco in [2] obtained the global-in-time existence and uniqueness of weak entropy solutions to the system (1.1) and analyzed the relaxation limits. In [33], recently Wang and Wang studied the initial value problem for the system (1.1) in multi-dimensions and obtained the pointwise estimates of classical solutions by using the method of Green function combined with some energy estimates. Also, the stability of planar rarefaction waves was discussed in [7,6]. Very recently, Ruan and Zhu [28] investigated the asymptotic decay rates toward the linear diffusion wave and also the rarefaction wave in R n , and proved the asymptotic relation of our Proposition 4.2 with k = 0 and the norm L 2 instead of H s−1−k in the case 2 ≤ n ≤ 7 by using the energy method and the semigroup argument.
In this paper we investigate the decay rate not only to the same linear diffusion wave as in [28] which can be seen from Proposition 4.2, but also to another diffusion wave which is given in terms of the heat kernel as shown in Theorem 2.4 for the initial value problem (1.1), (1.2) in R n for n ≥ 2 by applying the time-weighted energy method together with the semigroup argument , which removes the restriction n ≤ 7 assumed in [28] and also improves their results. To this end, we first transform the system (1.1) into the following equivalent decoupled system (1.4) which makes the derivation of our energy estimates easier, but is not essential for obtaining our main results, Then, by applying the time-weighted energy method to the decoupled system (1.4), we derive the optimal decay estimates of solutions for all n ≥ 1. Finally, using the semigroup argument, we show that the solution is asymptotic to the linear diffusion wave as t → +∞, provided that n ≥ 2. Our linear diffusion wave is given explicitly in terms of the heat kernel. The contents of the paper are as follows. In Section 2 we give full statements of our main theorems. Section 3 gives the proof of the results on the global existence and decay estimates of solutions. The last section gives the proof of the theorem on the asymptotic convergence to the linear diffusion wave.
Before closing this section, we give some notations to be used below. Let F [f ] denote the Fourier transform of f defined by and we denote its inverse transform by F −1 .
Let s be a nonnegative integer. Then H s = H s (R n ) denotes the Sobolev space of L 2 functions, equipped with the norm Here, for a nonnegative integer k, ∂ k x denotes the totality of all the k-th order derivatives with respect to x ∈ R n . Also, C k (I; H s (R n )) denotes the space of k-times continuously differentiable functions on the interval I with values in the Sobolev space H s = H s (R n ).
Finally, in this paper, we denote every positive constant by the same symbol C or c without confusion.

Main theorems
Our first theorem is on the global existence and uniform energy estimate of solutions to the problem (1.1), (1.2).
The next theorem is on the optimal decay estimates of solutions for initial data in H s (R n ) ∩ L 1 (R n ). Theorem 2.3. Let n ≥ 1, and let s ≥ 3 for n = 1 and s ≥ [ n 2 ] + 2 for n ≥ 2. Assume that u 0 ∈ H s (R n ) ∩ L 1 (R n ) and put E 1 := u 0 H s + u 0 L 1 . Then there exists a small positive constant δ 1 such that if E 1 ≤ δ 1 , then the global solution obtained in Theorem 2.1 satisfies the decay estimates Remark 1. To obtain the optimal decay estimates of solutions stated in Corollary 2.2 and Theorem 2.3, we will use the following decay estimate for L ∞ norm of the derivative ∂ x u as shown by N(T ) in (3.12): Our final result is concerning the asymptotic profile of the global solution obtained in Theorem 2.3 for n ≥ 2. First we show that for n ≥ 2, the solution to the problem (1.1), (1.2) can be approximated by the solution to the corresponding linear problem, Then we prove that the solution to this linear problem can be further approximated by the following simpler problem (2.7) based on the linear heat equation u t − ∆u = 0, Since the solution to the linear heat equation is asymptotic to the heat kernel we thus conclude that the asymptotic profile of our global solution is given by the following linear diffusion wave (u * , q * )(x, t): where M = R n u 0 (x)dx denotes the "mass". The result is precisely stated as follows.
Then the global solution (u, q) to the problem (1.1), (1.2), which is constructed in Theorem 2.3, is asymptotic to the linear diffusion wave (u * , q * ) in (2.9) as t → +∞: 3 Global existence and decay estimates

Global existence of solutions
This subsection is devoted to the proof of the global existence result stated in Theorem 2.1. 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. First we give a lemma which will be used in the derivation of our energy estimates.
Lemma 3.1. Let 1 ≤ p, q, r ≤ +∞ and 1 p = 1 q + 1 r . Then the following estimates hold: The proof of this lemma can be found in [20]. The next lemma shows the equivalence of the system (1. To get similar estimates for the derivatives, we apply ∂ l x to (3.1) 1 , obtaining ∇u. We multiply this equation by ∂ l x u and compute directly to get Integrating this equality with respect x and estimating the right hand side by applying Lemma 3.1, we obtain where 1 ≤ l ≤ s − 1. Here we have used the fact that f j (u) = O(u 2 ), j = 1, · · · , n for u → 0. We add (3.3) and (3.4) for 1 ≤ l ≤ s − 1 and integrate over (0, t). This yields where E 0 = u 0 H s . Letδ be a positive number (independnt of T ) and assume that sup 0≤t≤T u(t) H s ≤δ, where s ≥ [ n 2 ] + 2. Then the second term on the right hand side of (3.5) is estimated by Cδ t 0 ∂ x u(τ ) 2 H s−1 dτ . Therefore, choosingδ so small that Cδ ≤ 1 2 , we arrive at the uniform energy estimate On the other hand, it follows from (1.4) 2 that q H s+1 ≤ ∂ x u H s−1 , which combined with (3.6) yields These observations are summarized as follows.
By virtue of the a priori estimate (3.8) for small solutions stated in Proposition 3.3, we can apply the continuity argument and obtain a unique global solution to the problem (1.1), (1.2), provided that E 0 is suitably small, say, E 0 ≤ δ 0 . The solution obtained verifies (3.8) for t ≥ 0. This proves Theorem 2.1.

Optimal decay estimates
In this subsection, we obtain the optimal decay estimates of the solution constructed in Theorem 2.1 by using the time-weighted energy method. To this end, we define two time-weighted energy norms E(T ) and M(T ). Also, we introduce D(T ) as the dissipation norm corresponding to E(T ). (1 + t) where s ≥ 1. To derive estimates for E(T ), D(T ) and M(T ), we make use of the following time-weighted norm N(T ): As for the energy E(T ) and D(T ), we have the following estimate.
Proof. In order to prove this proposition, it is enough to show the following estimates for any t ∈ [0, T ] and 0 ≤ j ≤ s: (3.13) We know from (2.1) that (3.13) holds true for j = 0. Now, let 0 ≤ k ≤ s − 1 and suppose that (3.13) holds true for j = k. Then we show (3.13) for j = k + 1. Multiplying (3.4) by (1 + t) k+1 , integrating with respect to t over (0, t) and adding for l with k + 1 ≤ l ≤ s − 1, we have The second term on the right hand side is estimated by the induction hypothesis (3.13) with j = k, while the last term can be estimated by CN(T )D(T ) 2 . Consequently, we have 2 , which shows that (3.13) holds true for j = k + 1. Thus, by induction, we have proved Proposition 3.4.
By employing the optimal decay results expressed in E(T ) and M(T ), we obtain the following estimates for N(T ). Proof. (i) Let s 0 = [ n 2 ] + 1 and θ = n 2s 0 . By applying the Gagliardo-Nirenberg inequality, we see that Since n ≥ 2, it yields that N(T ) ≤ CE(T ).
(ii) By using the one-dimensional Gagliardo-Nirenberg inequality, we have Put X(T ) := E(T ) + D(T ). Then we have X(T ) 2 ≤ CE 2 0 + CX(T ) 3 . This inequality is solved as X(T ) ≤ CE 0 , provided that E 0 is suitably small. In particular, we have E(T ) ≤ CE 0 , which shows the desired decay estimate (2.2) for 0 ≤ k ≤ s. Moreover, by virtue of (1.4) 2 , we have To estimate the energy M(T ), we need the following L 1 estimate of the solution.
Lemma 3.6. Under the same assumptions as in Theorem 2.3, the solution to the problem (1.1), (1.2) satisfies the following L 1 estimate for u: (3.14) Proof. Applying (1 − ∆) −1 to (1.4) 1 , we have We denote by K(x) the the fundamental solution to the operator I − ∆, that is, (I −∆) −1 u = K * u. We know that K(x) ≥ 0, K ∈ L 1 and R n K(x)dx = 1. See [2] for the details. Let j δ be the Friedrichs mollifier and put We multiply (3.15) by φ δ (u) and integrate the resulting equation over R n × (0, t). Then, letting δ → 0, we obtain the desired L 1 estimate (3.14) just in the same way as in [2,16,28]. The details are omitted.
By employing the time-weighted energy method together with the L 1 estimate (3.14), we get the following estimate for M(T ). Proposition 3.7. Let n ≥ 1 and s ≥ [ n 2 ] + 2. Assume that u 0 ∈ H s (R n ) ∩ L 1 (R n ), and put E 0 := u 0 H s and E 1 := u 0 H s + u 0 L 1 . Then, if E 0 is suitably small, then the solution to the problem (1.1), (1.2) constructed in Theorem 2.1 satisfies following estimate: Proof. In order to prove this proposition, it is enough to show the following estimate for any t ∈ [0, T ] and 0 ≤ j ≤ s − 1: For this purpose, it is sufficient to prove for t ∈ [0, T ] and 0 ≤ j ≤ s − 1, where α > n 2 . First we show (3.17) for j = 0. We add (3.3) and (3.4) for 1 ≤ l ≤ s − 1, multiply the resulting inequality by (1 + t) α , and then integrate over (0, t). This yields (3.18) Since u(t) H s ≤ CE 0 by (2.1), we can estimate the term I 2 as where E 0 is assumed to be small as CE 0 ≤ 1 4 . On the other hand, we divide I 1 into two parts: To estimate I 12 , we choose T 1 so large that C(1 + T 1 ) −1 ≤ 1 4 . Then we divide the time interval [0, t] into two parts [0, T 1 ] and [T 1 , t]; here we treat the case t ≥ T 1 because the case t ≤ T 1 is easier. Thus we have where we have used (2.1). Finally, we estimate the term I 11 . By using the Gagliardo-Nirenberg inequality u L 2 ≤ C ∂ x u θ L 2 u 1−θ L 1 with θ = n n+2 and applying the Young inequality, we can estimate I 11 as where we have used the L 1 estimate (3.14) and the condition α > n 2 . Consequently, under the smallness assumption on E 0 , we arrive at the estimate which proves (3.17) for j = 0. Now, let 0 ≤ k ≤ s−2 and suppose that (3.17) holds true for j = k. Then we show (3.17) for j = k + 1. Multiplying (3.4) by (1 + t) k+1+α , integrating with respect to t over (0, t) and adding up for l with k + 1 ≤ l ≤ s − 1, we have Here, using the induction hypothesis (3.17) with j = k, we can estimate the second term on the right hand side by CE 2 1 (1 + N(T )) k (1 + t) α− n 2 . Similarly, we estimate the last term as Thus we obtain This shows that (3.17) holds true for j = k + 1. Thus, by induction, we have proved Proposition 3.7.
Proof of Theorem 2.3. By virtue of Lemma 3.5, we have N(T ) ≤ C(E(T ) + M(T )). Therefore we have from Propositions 3.4 and 3.7 that 3 . This inequality is solved as Y (T ) ≤ CE 1 , provided that E 1 is suitably small, say, E 1 ≤ δ 1 . In particular, we have M(T ) ≤ CE 1 , which proves the decay estimate (2.4) for 0 ≤ k ≤ s − 1. Moreover, using (1.4) 2 and (2.4) with k replaced by k + 1, we obtain This completes the proof of Theorem 2.3.

Asymptotic profile
The aim of this section is to prove Theorem 2.4 on the asymptotic profile.
To this end, we first consider the corresponding linear problem ( The solution operator G(t) * verifies the following decay property: Lemma 4.1. Let n ≥ 1 and s ≥ 0. If φ ∈ H s (R n ) ∩ L 1 (R n ), then we have the following decay estimate: Proof. By direct calculation, we have This completes the proof.
We decompose the solution formula (4.2) in the form u(t) =ū(t) − F (u)(t), whereū is the linear solution given in (4.1) and  Assume that u 0 ∈ H s (R n ) ∩ L 1 (R n ) and put E 1 := u 0 H s + u 0 L 1 . Let (u, q) be the global solution to the problem (1.1), (1.2) which is obtained in Theorem 2.1, and letū be the solution to the corresponding linear problem (2.6), which is given by the formula (4.1). Then we have for k with 0 ≤ k ≤ s − 1, where ρ(t) = ln(1 + t) for n = 2 and ρ(t) = 1 for n ≥ 3.
Proof. Let k and m be nonnegative integers. We apply ∂ k+m x to F (u) in (4.3) and take the L 2 norm, obtaining (4.4) By applying Lemma 4.1, using Lemma 3.1 and noticing that f j (u) = O(u 2 ), j = 1, · · · , n for u → 0, we have Similarly, we have We estimate the terms I 1 and I 2 . Let s 0 = [ n 2 ] + 1 and θ = n 2s 0 . By using the Gagliardo-Nirenberg inequality and (2.2), we see that (4.5) By using (2.2), (2.4) and (4.5), we estimate I 1 as for k satisfying k + m + 1 ≤ s, where ρ(t) is given above. Similarly, we can estimate I 2 as for k satisfying k + m + 1 ≤ s. We substitute (4.6) and (4.7) into (4.4) and take the sum for m with 0 ≤ m ≤ s − k − 1. This yields for k with 0 ≤ k ≤ s − 1. This completes the proof of Proposition 4.2.