Global Existence of Strong and Weak Solutions to the 2D Compressible Navier-Stokes-Poisson System in Bounded Domains with Large Data and Vacuum
Abstract
We study the two-dimensional compressible Navier-Stokes-Poisson system in a general simply connected bounded domain with Navier-slip boundary conditions and a positive, possibly non-flat, doping profile. We establish the global existence of a unique strong solution and of a global weak solution for arbitrarily large initial data allowing vacuum when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than 4/3. The main difficulties arise from the interaction between the Poisson force and the general boundary geometry. To overcome them, we introduce a modified material derivative and derive quantitative Lp estimates for the density by means of a Neumann duality argument together with a harmonic correction generated by the Navier-slip boundary condition. We further obtain a pointwise representation of the effective viscous flux by pulling back the Neumann Green function on the unit disk through a conformal mapping. The resulting commutator estimates, combined with elliptic estimates for the Poisson equation, yield a uniform-in-time upper bound for the density and allow us to close the higher-order a priori estimates. Finally, the global weak solution is constructed through a charge-preserving approximation and compactness argument.
Related
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- Existence and uniqueness of solutions for compressible magnetohydrodynamic equations with bounded density
- Global existence and uniqueness of Lions' weak solutions for the 2D inhomogeneous Navier-Stokes equations in bounded domains
- Existence and stability of shock profiles for a one-dimensional full compressible Navier--Stokes--Poisson equations
- Global well-posedness for the compressible Navier--Stokes equations in 3D bounded domains under a new smallness condition
BibTeX
@misc{du2026global,
title = {{Global Existence of Strong and Weak Solutions to the 2D Compressible Navier-Stokes-Poisson System in Bounded Domains with Large Data and Vacuum}},
author = {Yingzhi Du and Huijiang Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13618},
eprint = {2610.13618},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13618},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global Existence of Strong and Weak Solutions to the 2D Compressible Navier-Stokes-Poisson System in Bounded Domains with Large Data and Vacuum
AU - Yingzhi Du
AU - Huijiang Zhao
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13618
UR - https://arxiv.org/abs/2610.13618
AB - We study the two-dimensional compressible Navier-Stokes-Poisson system in a general simply connected bounded domain with Navier-slip boundary conditions and a positive, possibly non-flat, doping profile. We establish the global existence of a unique strong solution and of a global weak solution for arbitrarily large initial data allowing vacuum when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than 4/3. The main difficulties arise from the interaction between the Poisson force and the general boundary geometry. To overcome them, we introduce a modified material derivative and derive quantitative L^p estimates for the density by means of a Neumann duality argument together with a harmonic correction generated by the Navier-slip boundary condition. We further obtain a pointwise representation of the effective viscous flux by pulling back the Neumann Green function on the unit disk through a conformal mapping. The resulting commutator estimates, combined with elliptic estimates for the Poisson equation, yield a uniform-in-time upper bound for the density and allow us to close the higher-order a priori estimates. Finally, the global weak solution is constructed through a charge-preserving approximation and compactness argument.
ER -