Communications in Mathematical Sciences

Volume 14 (2016)

Number 5

Decay estimates of solutions to the compressible Navier–Stokes–Maxwell system in $\mathbb{R}^3$

Pages: 1189 – 1212

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n5.a1

Authors

Zhong Tan (School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, Fujian, China)

Leilei Tong (School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, Fujian, China)

Abstract

The compressible Navier–Stokes–Maxwell system with linear damping is investigated in $\mathbb{R}^3$, and the global existence and large-time behavior of solutions are established. We first construct the global unique solution under the assumptions that the $H^3$ norm of the initial data is small but that the higher-order derivatives can be arbitrarily large. Further, if the initial data belongs to $\dot{H}^{-s} (0 \leq s \lt 3/2)$ or $\dot{B}^{-s}_{2,\infty} (0 \lt s \leq 3/2)$, by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher-order derivatives. As an immediate by-product, the $L^p - L^2 (1 \leq p \leq 2)$ type of the decay rates follow without requiring that the $L^p$ norm of initial data is small.

Keywords

compressible Navier–Stokes–Maxwell system, global solution, time decay rate, energy method, interpolation

2010 Mathematics Subject Classification

35B40, 35Q30, 35Q35, 35Q61, 76N10, 82D37

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Published 18 May 2016