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 $L^{q_0}$ 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.