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# 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

#### 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

Published 18 May 2016