A uniqueness criterion for viscous limits of boundary Riemann problems

TL;DR

This paper establishes conditions under which different viscous approximations of boundary Riemann problems for conservation laws yield the same limit, enhancing understanding of viscous limits in boundary value problems.

Contribution

It provides a new uniqueness criterion for viscous limits, applicable to both characteristic and non-characteristic cases without requiring genuine nonlinearity or linear degeneracy.

Findings

Different viscous approximations can converge to the same limit under certain conditions.

The self-similar second-order and classical viscous approximations share the same limit.

The analysis applies broadly, without restrictions on the nature of characteristic fields.

Abstract

We deal with initial-boundary value problems for systems of conservation laws in one space dimension and we focus on the boundary Riemann problem. It is known that, in general, different viscous approximations provide different limits. In this paper, we establish sufficient conditions to conclude that two different approximations lead to the same limit. As an application of this result, we show that, under reasonable assumptions, the self-similar second-order approximation and the classical viscous approximation provide the same limit. Our analysis applies to both the characteristic and the non characteristic case. We require neither genuine nonlinearity nor linear degeneracy of the characteristic fields.