Global Well-posedness of the free boundary problem for the compressible Euler equations with damping and gravity

Abstract

We consider the free boundary problem for a compressible barotropic fluid lying above a rigid bottom and below the air, in a horizontally periodic setting. The fluid dynamics is governed by the compressible Euler equations with damping and gravity, and the effect of surface tension is neglected on the upper free surface. We prove the global well-posedness of the problem for the initial data around the equilibrium and that the solution decays to the equilibrium at an almost exponential rate. The difficulties caused by the less regularity of the free surface and the troublesome linear effects induced by the nontrivial equilibrium density are overcome by using certain good unknowns.