On Vacuum Free Boundary Problem of the Spherically Symmetric Euler Equations with Damping and Solid Core

TL;DR

This paper proves the global existence and exponential decay to equilibrium of smooth solutions for a vacuum free boundary problem in spherically symmetric Euler equations with damping and a solid core, under certain conditions.

Contribution

It establishes the first rigorous proof of global smooth solutions and exponential convergence for this class of free boundary Euler problems with damping and solid core.

Findings

Global existence of smooth solutions is proven.

Solutions converge exponentially to equilibrium.

Results hold for arbitrary positive gas constant and small initial perturbations.

Abstract

In this paper, the global existence of smooth solution and the long-time asymptotic stability of the equilibrium to vacuum free boundary problem of the spherically symmetric Euler equations with damping and solid core have been obtained for arbitrary finite positive gas constant $A$ in the state equation $p=A \rho^\gamma$ with $p$ being the pressure and $\rho$ the density, provided that $\gamma>4/3,$ initial perturbation is small and the radius of the equilibrium $R$ is suitably larger than the radius of the solid core $r_0$. Moreover, we obtain the pointwise convergence from the smooth solution to the equilibrium in a surprisingly exponential time-decay rate. The proof is mainly based on weighted energy method in Lagrangian coordinate.