On Spherically Symmetric Motions of the Atmosphere Surrounding a Planet Governed by the Compressible Euler Equations

TL;DR

This paper studies spherically symmetric motions of compressible gas around a planet, addressing equilibrium states, linearized solutions, and overcoming free boundary singularities using the Nash-Moser theorem.

Contribution

It constructs true solutions near equilibrium states with time-periodic behavior, handling free boundary singularities via Nash-Moser techniques.

Findings

Existence of equilibrium states touching vacuum with finite radii

Linearized equations admit time-periodic solutions

Successful application of Nash-Moser theorem to free boundary problems

Abstract

We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.