Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

Stability of the planar rarefaction wave to three-dimensional compressible model of viscous ions motion

Pages: 1735 – 1762



Yeping Li (School of Sciences, Nantong University, Nantong, China)

Zhen Luo (School of Mathematical Sciences, Xiamen University, Xiamen, China)

Jiahong Wu (Department of Mathematics, Oklahoma State University, Stillwater, Ok., U.S.A.)


The compressible Navier–Stokes–Poisson equations model the motion of viscous ions and play important roles in the study of self-gravitational viscous gaseous stars and in the simulations of charged particles in semiconductor devices and plasmas physics. This paper establishes the stability and precise large-time behavior of perturbations near the planar rarefaction wave to three-dimensional isentropic compressible Navier–Stokes–Poisson equations. The results presented in this paper are new. Previous studies focused on the one-dimensional compressible Navier–Stokes–Poisson equations and little has been done for the multi-dimensional case. In order to prove the desired asymptotic stability, we take into account both the effect of the self-consistent electrostatic potential and the decay rate of the planar rarefaction wave. Due to the complexity of the nonlinearity and the effect of the self-consistent electric field, the proof involves highly non-trivial a priori bounds.


Navier–Stokes–Poisson equations, planar rarefaction wave, stability

2010 Mathematics Subject Classification

35B35, 35B40, 76N15

Y. Li is supported in part by the National Science Foundation of China (Grant No. 12171258).

Z. Luo is supported in part by the National Science Foundation of China (Grant No. 12171401) and the Natural Science Foundation of Fujian Province of China (Grant No. 2020J01029).

J. Wu is partially supported by NSF grant DMS 1624146 and the AT&T Foundation at Oklahoma State University.

Received 2 December 2020

Received revised 29 December 2021

Accepted 21 January 2022

Published 14 September 2022