Global Well-posedness of Compressible Viscous Surface Waves without Surface Tension

TL;DR

This paper proves the global well-posedness of the compressible viscous surface wave problem without surface tension, using new control of spatial derivatives and time-weighted energy estimates, in both 2D and 3D.

Contribution

It establishes the first global well-posedness result for compressible viscous surface waves without surface tension, employing novel control of Eulerian derivatives and nonlinear cancellations.

Findings

Global well-posedness in 2D and 3D

No low frequency assumption on initial data

New control of Eulerian spatial derivatives

Abstract

We consider the free boundary problem for a layer of compressible viscous barotropic fluid lying above a fixed rigid bottom and below the atmosphere of positive constant pressure. The fluid dynamics is governed by the compressible Navier--Stokes equations with gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the reformulated problem in flattening coordinates near the equilibrium in both two and three dimensions without any low frequency assumption of the initial data. The key ingredients here are the new control of the {\it Eulerian spatial derivatives} of the solution, which benefits a crucial nonlinear cancellation of the highest order spatial regularity of the free boundary, and the time weighted energy estimates.