Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$

Zhong Tan; Yong Wang; Xu Zhang

doi:10.3934/krm.2012.5.615

Article Contents

2012, Volume 5, Issue 3: 615-638. Doi: 10.3934/krm.2012.5.615

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Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$

1.
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005

Received: December 31, 2011

Revised: January 31, 2012

Published: August 2012

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Abstract

We are concerned with the long-time behavior of global strong solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$, where the electric field is governed by the self-consistent Poisson equation. When the regular initial perturbations belong to $H^{4}(\mathbb{R}^{3})\cap \dot{B}_{1,\infty}^{-s}(\mathbb{R}^{3})$ with $s\in [0,1]$, we show that the density and momentum of the system converge to their equilibrium state at the optimal $L^2$-rates $(1+t)^{-\frac{3}{4}-\frac{s}{2}}$ and $(1+t)^{-\frac{1}{4}-\frac{s}{2}}$ respectively, and the decay rate is still $(1+t)^{-\frac{3}{4}}$ for temperature which is proved to be not optimal.

Keywords:

Compressible Navier-Stokes-Poisson system,
optimal decay rates.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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