Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system

TL;DR

This paper establishes the uniform stability of constant equilibria in the 3D Navier-Stokes-Poisson system as viscosity vanishes, combining parabolic and dispersive techniques to prove global solutions and decay rates.

Contribution

It introduces a novel approach that merges parabolic energy estimates with dispersive techniques to analyze the inviscid limit of the Navier-Stokes-Poisson system.

Findings

Uniform stability of equilibria in the inviscid limit

Existence of unique global smooth solutions

Time decay rates independent of viscosity parameter

Abstract

We prove a stability result of constant equilibria for the three-dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the incompressible part of the initial velocity is assumed to be small compared to $\varepsilon$. We then get a unique global smooth solution. We also prove a uniform in $\varepsilon$ time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at $\varepsilon$ fixed and the dispersive techniques (dispersive estimates and normal form transformation) that are useful for the inviscid irrotational system.