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Stability of planar shock wave for the 3-dimensional compressible Navier-Stokes-Poisson equations
Authors
Xiaochun Wu
Publication date
17 August 2023
Publisher
View
on
arXiv
Abstract
This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain
Ω
:
=
R
×
T
2
\Omega:=\mathbb{R}\times \mathbb{T}^2
Ω
:=
R
×
T
2
with
T
2
=
(
R
/
Z
)
2
\mathbb{T}^2=(\mathbb{R}/\mathbb{Z})^2
T
2
=
(
R
/
Z
)
2
. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below
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oai:arXiv.org:2308.08838
Last time updated on 24/08/2023