Communications on Pure and Applied Analysis (CPAA)

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

Pages: 2283 - 2313, Volume 14, Issue 6, November 2015      doi:10.3934/cpaa.2015.14.2283

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Weike Wang - Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai, China (email)
Xin Xu - Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R.China, China (email)

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.

Received: January 2015;      Revised: July 2015;      Available Online: September 2015.