Large time behavior of solution for the full compressible navier-stokes-maxwell system

Weike Wang; Xin  Xu

doi:10.3934/cpaa.2015.14.2283

Article Contents

2015, Volume 14, Issue 6: 2283-2313. Doi: 10.3934/cpaa.2015.14.2283

This issue Previous Article On Fractional Schrödinger Equations in sobolev spaces Next Article Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

Large time behavior of solution for the full compressible navier-stokes-maxwell system

Weike Wang^1, and
Xin Xu^2,

1.
Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai
2.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R.China

Received: December 31, 2014

Revised: June 30, 2015

Published: October 2015

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Abstract

In this paper, the Cauchy problem for the compressible Navier-Stokes-Maxwell equation is studied in $R^3$, the $L^p$ time decay rate for the global smooth solution is established. Our method is mainly based on a detailed analysis to the Green's function of the linearized system and some elaborate energy estimates. To give the explicit representation of the Green's function, we use the Helmholtz decomposition by which we can decompose the solution into two parts and give the expression to each part. Our results show a sharp difference between the decay of solution for Navier-Stokes-Maxwell system and that for the Navier-Stokes equation.

Keywords:

Full compressible Navier-Stokes-Maxwell system,
Green's function,
Helmholtz decomposition,
energy estimate,
time decay estimate.

Mathematics Subject Classification: 35J08, 35B40, 35Q30, 35Q61.

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