Propagating fronts for a viscous Hamer-type system

Abstract

Motivated by radiation hydrodynamics, we analyse a \begin{document}$ 2\times2 $\end{document} 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 [21]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.