Asymptotic behavior of solutions to a hyperbolic–elliptic coupled system in multi-dimensional radiating gas

Abstract

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem of a hyperbolic-elliptic coupled system in the multi-dimensional radiating gasFirst, for the case with the same end states u − = u + = 0, we prove the existence and uniqueness of the global solutions to the above Cauchy problem by combining some a priori estimates and the local existence based on the continuity argument. Then L p -convergence rates of solutions are respectively obtained by applying L 2 -energy method for n = 1, 2, 3 and L p -energy method for 3 < n < 8 and interpolation inequality. Furthermore, by semigroup argument, we obtain the decay rates to the diffusion waves for 1 n < 8. Secondly, for the case with the different end states u − < u + , our main concern is that the corresponding Cauchy problem in ndimensional space (n = 1, 2, 3) behaviors like planar rarefaction waves. Its convergence rate is also obtained by L 2 -energy method and L 1 -estimate.

Asymptotic behavior of solutions to a model system of a radiating gas

“…Then, letting δ → 0, we obtain the desired L 1 estimate (29) just in the same way as in [2,16,29]. Then, letting δ → 0, we obtain the desired L 1 estimate (29) just in the same way as in [2,16,29].…”

“…Then, by applying the time-weighted energy method to the decoupled system (4), we derive the optimal decay estimate of solutions for all n ≥ 1 (see Theorem 2.2), which is also an improvement on the corresponding result in [29]. Then, by applying the time-weighted energy method to the decoupled system (4), we derive the optimal decay estimate of solutions for all n ≥ 1 (see Theorem 2.2), which is also an improvement on the corresponding result in [29].…”

“…Recently, in [34], Wang and Wang studied the initial value problem for the system (1) in multi-dimensions and obtained the pointwise estimates of classical solutions by using the method of Green function combined with some energy estimates. Very recently, Ruan and Zhu [29] investigated the asymptotic behavior of solutions to (1) in R n with n ≤ 7. By using the energy method and the semigroup argument, they obtained decay rates toward a solution to the linearized problem (10) under the restriction 2 ≤ n ≤ 7.…”

“…Very recently, Ruan and Zhu [29] investigated the asymptotic behavior of solutions to (1) in R n with n ≤ 7. Concerning the stability of planar rarefaction waves, we refer to [7,6,29]. Concerning the stability of planar rarefaction waves, we refer to [7,6,29].…”

“…In this paper we investigate the decay rate to the linear diffusion wave for the initial value problem (1), (2) in R n for n ≥ 2 by applying the time-weighted energy method together with the semigroup argument and remove the restriction n ≤ 7 assumed in [29] (see Theorem 2.3). Our linear diffusion wave is simple enough and is given explicitly in terms of the heat kernel (see (13)), so that our result is an improvement on the previous one in [29].…”