Hydrostatic bubbles of compressible fluid in an incompressible fluid

TL;DR

This paper investigates stationary configurations of compressible bubbles within an incompressible fluid under gravity, developing bifurcation methods and functional analysis tools to construct nearly spherical solutions.

Contribution

It introduces a bifurcation approach to construct nontrivial gravity-affected solutions and develops new functional analytic tools for weighted Sobolev spaces.

Findings

Existence of smooth, nearly spherical solutions with gravity

Development of linear and nonlinear functional analysis tools

Bifurcation methods applied to fluid interface problems

Abstract

We study stationary configurations of compressible barotropic fluids lying inside an incompressible fluid and acted upon by a constant gravitational field. Without gravity, it is a simple matter to construct solutions consisting of perfectly spherical bubbles of compressible fluid, but with gravity the stratification of the compressible fluid's density prevents the existence of solutions with such simple geometries. We construct smooth solutions with nontrivial gravity and nearly spherical, axisymmetric geometry by means of a bifurcation argument in an appropriate weighted Sobolev space, defined in terms of the Legendre operator. Along the way, we develop a number of essential linear and nonlinear functional analytic tools for this scale of spaces, which are of independent interest.