Global solutions to the compressible Navier-Stokes-Poisson equations in the half-space

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 $H^3$, 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 $\frac12\leq s<1$, yield the rate $(1+t)^{-s/2}$. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for $0<s<\frac14$. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.