Almost global existence and nonlinear asymptotic stability for bubble dynamics in inviscid compressible liquid

Abstract: The present paper considers the full nonlinear dynamics of a homogeneous bubble inside an unbounded isentropic compressible inviscid liquid. This model is described by a free-boundary problem of compressible Euler equations with nonlinear boundary conditions. The liquid is governed by the compressible Euler equation, while the bubble surface is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. This classical model is of great concern in physics due to its wide applications. We begin by proving the local existence and uniqueness using energy methods under an iteration scheme. For long-time behavior, we developed a generalized weighted space-time estimate, which extends the Keel-Smith-Sogge estimate to nonlinear wave equations regardless of the boundary conditions, at the cost of the appearance of a boundary term with only lowest-order derivatives. This term is handled by using characteristics to track the backward pressure wave. Then the almost global existence and nonlinear radiative decay are proved through a bootstrap argument, which encompasses the energy estimate, the generalized Keel-Smith-Sogge estimate, and the analysis of backward pressure waves. The analysis of the backward pressure wave by characteristics involves a loss of derivative due to the quasilinear nature of the system. This is overcome by the above generalized weighted spacetime estimate with the lowest-order boundary term, which actually provides a mechanism to gain derivatives back. The coupling of these two methods is the novelty of the present paper and can not only be used for the current question but is expected to be applied to other questions regarding nonlinear wave equations with complicated boundary conditions.