Vanishing viscosity limits of compressible viscoelastic equations in half space

TL;DR

This paper investigates the vanishing viscosity limit of solutions to three-dimensional compressible viscoelastic equations in a half space, establishing uniform regularity estimates and addressing boundary conditions to justify the limit.

Contribution

It provides the first uniform regularity estimates for compressible viscoelastic equations in a half space and justifies the vanishing viscosity limit using Lagrangian coordinates.

Findings

Uniform regularity estimates established for solutions.

Boundary layer effects are mitigated by the deformation gradient.

Different boundary conditions are suitable for well-posedness via the vanishing viscosity limit.

Abstract

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, these results also provide two different kinds of suitable boundary conditions for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the vanishing viscosity limit method. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper.