Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
Abstract
We construct global weak solutions to the isentropic compressible Navier-Stokes and Navier-Stokes-Poisson systems in the unit disk under no-slip boundary conditions, allowing vacuum and unbounded densities for sufficiently small initial energy. Such solutions particularly possess uniform Lq0 density bounds for some q0>4γ, beyond the natural energy class. Our analysis combines Green function representations with the mean-value property of harmonic functions to isolate a pressure-generated harmonic component of the effective viscous flux and obtain a purely interior potential representation valid for both systems. These results complement Perepelitsa's half-space theory (Arch. Ration. Mech. Anal. 212 (2014), pp. 709-726) and our recent half-plane work (arXiv:2601.11852, 2026), treating bounded-domain geometry and Poisson coupling with nonnegative initial density.
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BibTeX
@misc{wang2026global,
title = {{Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions}},
author = {Shuai Wang and Xin Zhong},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01792},
eprint = {2611.01792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01792},
}
RIS
TY - UNPB
TI - Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
AU - Shuai Wang
AU - Xin Zhong
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01792
UR - https://arxiv.org/abs/2611.01792
AB - We construct global weak solutions to the isentropic compressible Navier-Stokes and Navier-Stokes-Poisson systems in the unit disk under no-slip boundary conditions, allowing vacuum and unbounded densities for sufficiently small initial energy. Such solutions particularly possess uniform L^q_0 density bounds for some q_0>4γ, beyond the natural energy class. Our analysis combines Green function representations with the mean-value property of harmonic functions to isolate a pressure-generated harmonic component of the effective viscous flux and obtain a purely interior potential representation valid for both systems. These results complement Perepelitsa's half-space theory (Arch. Ration. Mech. Anal. 212 (2014), pp. 709-726) and our recent half-plane work (arXiv:2601.11852, 2026), treating bounded-domain geometry and Poisson coupling with nonnegative initial density.
ER -