Vanishing viscosity limits for the free boundary problem of compressible viscoelastic fluids with surface tension

TL;DR

This paper analyzes the vanishing viscosity limit for free boundary problems in compressible viscoelastic fluids with surface tension, showing the boundary layer does not form, unlike in classical Navier-Stokes systems.

Contribution

It establishes uniform regularity of classical solutions in Sobolev spaces and demonstrates the absence of boundary layers in the vanishing viscosity limit for viscoelastic fluids.

Findings

Uniform regularity in Sobolev spaces for solutions

No boundary layer forms in the limit

Difference from classical Navier-Stokes results

Abstract

We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate regularities of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids which is different to the case studied by the second author for the free boundary compressible Navier-Stokes system.