Characteristic boundary layers for mixed hyperbolic-parabolic systems in one space dimension, and applications to the Navier-Stokes and MHD equations

TL;DR

This paper analyzes boundary layers in mixed hyperbolic-parabolic systems in one dimension, focusing on the zero viscosity limit and applications to Navier-Stokes and MHD equations, especially in challenging characteristic boundary cases.

Contribution

It provides a detailed analysis of boundary layers and the boundary Riemann problem for mixed systems, including the doubly characteristic case, applicable to Navier-Stokes and MHD equations.

Findings

Characterization of boundary layers in mixed hyperbolic-parabolic systems.

Solution description of the boundary Riemann problem as viscosity approaches zero.

Extension of analysis to non-conservative systems and challenging characteristic cases.

Abstract

We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null conductivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to non-conservative systems.