Nonlinear Stability of Linearly Expanding Goldreich-Weber Solutions for the Navier-Stokes-Poisson System with Degenerate Viscosity Under Radial Perturbations

Abstract

In this work, we study the nonlinear stability of linearly expanding Goldreich-Weber (GW) solutions for the gravitational Navier-Stokes-Poisson system with pressure law $p(\ddot{I})=\ddot{I}{}^{\frac{4}{3}}$ and degenerate viscosity. It is well-known that linearly expanding GW solutions are special solutions to the Euler-Poisson system with ${\hat{I}}^3=\frac{4}{3}$. With bulk viscosity equal to 0, linearly expanding GW solutions are also solutions to the Navier-Stokes-Poisson equations. Choosing shear viscosity proportional to $\ddot{I}{}^{\hat{I}}\pm$ with $0<\hat{I}\pm \le \frac{2}{3}$ and zero bulk viscosity, we prove the nonlinear stability of linearly expanding GW solutions under radial perturbation.