Communications in Mathematical Sciences

Volume 22 (2024)

Number 3

Global existence and asymptotic behavior of the full Euler system with damping and radiative effects in $\mathbb{R}^3$

Pages: 789 – 816

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n3.a8

Authors

Shijin Deng (School of Mathematical Sciences, CMA, Shanghai Jiao Tong University, Shanghai, China)

Wenjun Wang (College of Science, University of Shanghai for Science and Technology, Shanghai, China)

Feng Xie (School of Mathematical Sciences, Shanghai, Shanghai Jiao Tong University, Shanghai, China)

Xiongfeng Yang (School of Mathematical Sciences, Shanghai, Shanghai Jiao Tong University, Shanghai, China)

Abstract

In this paper, we study the global existence and the large-time behavior of solutions to the Cauchy problem of the full Euler system with damping and radiative effects around some constant equilibrium states. It is well-known that the solutions may blow up in finite time without the additional damping and radiative effects, and the global existence of the solutions obtained in this paper shows that these two effects together prevent the formation of the singularity when the initial perturbation is small. Combining the Green’s function method and energy estimates, we consider the pointwise structures of the solutions to obtain a precise description of the system. The construction of the Green’s function includes three steps: singularity removal, long wave-short wave decomposition and weighted energy estimate. Finally, we achieve the pointwise estimates of the solutions in the small perturbation framework by Duhamel’s principle, the pointwise structure of the Green’s function established for the linearized equations and bounded estimates for higher order derivatives of the solutions together.

Keywords

full Euler system with damping and radiative effects, global existence, pointwise estimates, Green’s function

2010 Mathematics Subject Classification

35L60, 35M31, 35Q35, 76N10

Full Text (PDF format)

Received 18 March 2023

Received revised 6 August 2023

Accepted 1 September 2023

Published 4 March 2024