Coupling techniques for nonlinear hyperbolic equations. II. Resonant interfaces with internal structure

TL;DR

This paper analyzes the internal structure of interfaces in coupled nonlinear hyperbolic equations, deriving criteria for solution selection and exploring non-uniqueness in non-convex flux scenarios.

Contribution

It introduces a new selection criterion for interface solutions and characterizes the nonconservative interface layer in coupled hyperbolic equations.

Findings

Derived a selection criterion linked to regularization mechanisms.

Characterized the nonconservative interface layer.

Provided evidence of non-uniqueness for non-convex flux functions.

Abstract

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the {restricted} case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.