Uniform regularity for the free surface compressible Navier-Stokes equations with or without surface tension

TL;DR

This paper establishes uniform regularity results for solutions to the 3D compressible Navier-Stokes equations with free surfaces, analyzing their behavior as viscosity and surface tension vanish, and proving convergence to the Euler system.

Contribution

It provides the first uniform bounds for solutions independent of viscosity and surface tension, and rigorously derives the asymptotic limits to the Euler equations.

Findings

Existence of unique strong solutions independent of viscosity and surface tension.

Uniform boundedness in $W^{1, ext{infinity}}$ and conormal Sobolev spaces.

Boundary layer for density is weaker than for velocity.

Abstract

In this paper, we investigate the uniform regularity of solutions to the 3-dimensional isentropic compressible Navier-Stokes system with free surfaces and study the corresponding asymptotic limits of such solutions to that of the compressible Euler system for vanishing viscosity and surface tension. It is shown that there exists an unique strong solution to the free boundary problem for the compressible Navier-Stokes system in a finite time interval which is independent of the viscosity and the surface tension. The solution is uniform bounded both in $W^{1,\infty}$ and a conormal Sobolev space. It is also shown that the boundary layer for the density is weaker than the one for the velocity field. Based on such uniform estimates, the asymptotic limits to the free boundary problem for the ideal compressible Euler system with or without surface tension as both the viscosity and the surface tension tend to zero, are established by a strong convergence argument.