Global solutions to the compressible Navier--Stokes--Poisson equations in the half-space and exterior domains
Abstract
We study the initial boundary value problem for the compressible Navier--Stokes--Poisson equations without damping in the three-dimensional half-space and in smooth exterior domains. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in H3, with a small initial electric field in L2, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy expressed through the Helmholtz projection. In exterior domains, boundary normal coordinates and localized elliptic estimates control the geometric terms. We also obtain algebraic decay in the full energy norm. In the half-space, finite tangential negative Sobolev norms of order −s, with 1/2≤ s<1, yield the rate (1+t)−s/2. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for 0<s<1/4. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.
Related
- An Lp-theory for global weak solutions to the Navier-Stokes equations in exterior domains
- Global Existence of Strong and Weak Solutions to the 2D Compressible Navier-Stokes-Poisson System in Bounded Domains with Large Data and Vacuum
- Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
- Asymptotic behavior of generic Riemann solutions to one-dimensional full compressible Navier-Stokes-Poisson equations
- Large time behavior of global solutions to one-dimensional non-isentropic compressible Navier-Stokes-Poisson equations with large initial data
BibTeX
@misc{liu2026global,
title = {{Global solutions to the compressible Navier--Stokes--Poisson equations in the half-space and exterior domains}},
author = {Hairong Liu and Jinlong Wu and Hua Zhong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15588},
eprint = {2610.15588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15588},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global solutions to the compressible Navier--Stokes--Poisson equations in the half-space and exterior domains
AU - Hairong Liu
AU - Jinlong Wu
AU - Hua Zhong
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.15588
UR - https://arxiv.org/abs/2610.15588
AB - We study the initial boundary value problem for the compressible Navier--Stokes--Poisson equations without damping in the three-dimensional half-space and in smooth exterior domains. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in H^3, with a small initial electric field in L^2, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy expressed through the Helmholtz projection. In exterior domains, boundary normal coordinates and localized elliptic estimates control the geometric terms. We also obtain algebraic decay in the full energy norm. In the half-space, finite tangential negative Sobolev norms of order −s, with 1/2≤ s<1, yield the rate (1+t)^−s/2. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for 0<s<1/4. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.
ER -