Propagating Fronts for a Viscous Hamer-Type system

TL;DR

This paper proves the existence of smooth propagating fronts in a viscous Hamer-type system, which regularize shocks in a viscous conservation law coupled with an elliptic equation, using Geometric Singular Perturbation Theory.

Contribution

It introduces a rigorous analysis of shock regularization in a viscous coupled system, extending the understanding of viscous effects in radiation hydrodynamics models.

Findings

Existence of smooth propagating fronts for small viscosity and large shocks.

Regularization of inviscid shock discontinuities by viscous fronts.

Application of Geometric Singular Perturbation Theory to coupled PDE systems.

Abstract

Motivated by radiation hydrodynamics, we analyse a 2x2 system consisting of a one-dimensional viscous conservation law with strictly convex flux -- the viscous Burgers' equation being a paradigmatic example -- coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity -- usually called sub-shock -- it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on Geometric Singular Perturbation Theory (GSPT) as introduced in the pioneering work of Fenichel [5] and subsequently developed by Szmolyan [19]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.