Stability of Gaseous Stars in Spherically Symmetric Motions

Abstract

We study the linearized stability of stationary solutions of gaseous stars which are in spherically symmetric and isentropic motion. If viscosity is ignored, we have following three types of problems: (EC), Euler equation with a solid core; (EP), Euler--Poisson equation without a solid core; (EPC), Euler--Poisson equation with a solid core. In Lagrangian formulation, we prove that any solution of (EC) is neutrally stable. Any solution of (EP) and (EPC) is also neutrally stable when the adiabatic index $\gamma \in (\frac{4}{3}, 2)$ and unstable for (EP) when $\gamma \in (1, \frac{4}{3})$. Moreover, for (EPC) and $\gamma \in$ (1, 2), any solution with small total mass is also neutrally stable. When viscosity is present $(\nu > 0)$, the velocity disturbance on the outer surface of gas is important. For $\nu > 0$, we prove that the neutrally stable solution (when $\nu = 0$) is now stable with respect to positive-type disturbances, which include Dirichlet and Neumann boundary conditions. The solution can be unstable with respect to disturbances of some other types. The problems were studied through spectral analysis of the linearized operators with singularities at the endpoints of intervals.