Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces

TL;DR

This paper explores analytical and numerical methods for coupling nonlinear hyperbolic systems, focusing on interface dynamics, resonance effects, and establishing existence results for Riemann problems with complex wave interactions.

Contribution

It introduces an augmented formulation for interface modeling and provides nonlinear wave interaction estimates applicable to resonant wave patterns in hyperbolic systems.

Findings

Existence theorem for Riemann problem under general assumptions

Wave interaction estimates for resonant patterns

Analysis of non-uniqueness and admissibility conditions

Abstract

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce here an augmented formulation which allows for the modeling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial value problem which need to be supplemented with further admissibility conditions. This first paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions which apply to resonant wave patterns.