Existence and stability of shock profiles for a one-dimensional full compressible Navier--Stokes--Poisson equations
Abstract
This paper studies the existence and asymptotic stability of traveling shock waves for the one-dimensional full compressible Navier--Stokes--Poisson equations, which model viscous heat-conducting ions in a self-consistent electrostatic field. We first prove the existence and uniqueness of a smooth shock profile of small amplitude by the center manifold theorem, and then establish its large-time asymptotic stability under suitably small smooth perturbations. The proof is based on an energy estimate for a specially constructed integrated system derived from the full compressible Navier--Stokes--Poisson equations. To carry out this estimate, we introduce suitable anti-derivative variables that incorporate the effects of the self-consistent electrostatic potential and heat conduction, thereby reformulating the original system as a dissipative integrated one. The subsequent energy estimates are completed by carefully handling the induced couplings.
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BibTeX
@misc{li2026existence,
title = {{Existence and stability of shock profiles for a one-dimensional full compressible Navier--Stokes--Poisson equations}},
author = {Yeping Li and Yuan Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04257},
eprint = {2610.04257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04257},
}
RIS
TY - UNPB
TI - Existence and stability of shock profiles for a one-dimensional full compressible Navier--Stokes--Poisson equations
AU - Yeping Li
AU - Yuan Yuan
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04257
UR - https://arxiv.org/abs/2610.04257
AB - This paper studies the existence and asymptotic stability of traveling shock waves for the one-dimensional full compressible Navier--Stokes--Poisson equations, which model viscous heat-conducting ions in a self-consistent electrostatic field. We first prove the existence and uniqueness of a smooth shock profile of small amplitude by the center manifold theorem, and then establish its large-time asymptotic stability under suitably small smooth perturbations. The proof is based on an energy estimate for a specially constructed integrated system derived from the full compressible Navier--Stokes--Poisson equations. To carry out this estimate, we introduce suitable anti-derivative variables that incorporate the effects of the self-consistent electrostatic potential and heat conduction, thereby reformulating the original system as a dissipative integrated one. The subsequent energy estimates are completed by carefully handling the induced couplings.
ER -