Boundary layers for self-similar viscous approximations of nonlinear hyperbolic systems

TL;DR

This paper characterizes residual boundary conditions from self-similar viscous approximations of nonlinear hyperbolic systems and shows their limits coincide with classical vanishing viscosity solutions under broad conditions.

Contribution

It provides a detailed description of boundary conditions for self-similar viscous approximations and establishes their convergence to classical solutions without requiring nonlinearity or degeneracy.

Findings

Residual boundary conditions are precisely characterized.

Limits of self-similar and classical vanishing viscosity approximations coincide.

Results apply to both conservative and non-conservative systems.

Abstract

We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and non conservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishing viscosity approximation coincide. We require neither genuinely nonlinearity nor linear degeneracy of the characteristic fields.