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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 7

### Long-time behavior of solutions to the non-isentropic Euler–Poisson system in $\mathbb{R}^3$

Pages: 1947 – 1965

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n7.a8

#### Authors

#### Abstract

We study the global existence and asymptotic behavior of smooth solutions near a nonflat steady state to the compressible non-isentropic Euler–Poisson system in $\mathbb{R}^3$. Using some concise energy estimates and an interpolation trick, we show that the solution converges to the stationary solution exponentially fast. Here our results can follow from that the $H^3$ norms of the initial density, velocity and temperature are small. In this sense, we reduce the regularity of the initial temperature in [Y.P. Li, J. Differential Equations, 225:134–167, 2006].

#### Keywords

non-isentropic Euler–Poisson system, long-time behavior, energy method, interpolation

#### 2010 Mathematics Subject Classification

35B40, 35M10, 35Q35, 35Q60, 76N10

The work of the first author was partially supported by the National Natural Science Foundation of China (No. 11501143) and the Science and Technology Foundation of Guizhou Province of China (Nos. LKS[2012]11, LKS[2013]05), and the PhD launch scientific research projects of Guizhou Normal University (No. 2014). The work of the second author was partially supported by the National Natural Science Foundation of China (Nos. 11271305, 11531010). The work of the third author was partially supported by the National Natural Science Foundation of China (No. 11701264) and SUSTC startup fund (No. 28/Y01286211).

Received 29 March 2017

Accepted 13 June 2017

Published 16 October 2017