Propagating fronts for a viscous Hamer-type system

Abstract

<p style='text-indent:20px;'>Motivated by radiation hydrodynamics, we analyse a <inline-formula><tex-math id="M1">\begin{document}$ 2\times2 $\end{document}</tex-math></inline-formula> 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 <b>viscous Hamer-type system</b>. 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 <i>sub-shock</i>– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on <i>Geometric Singular Perturbation Theory</i> (GSPT) as introduced in the pioneering work of Fenichel [<xref ref-type="bibr" rid="b5">5</xref>] and subsequently developed by Szmolyan [<xref ref-type="bibr" rid="b21">21</xref>]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.</p>