Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law

TL;DR

This paper proves the long-term stability of a combined shock and rarefaction wave in a non-convex conservation law with Cattaneo's heat flux law, under small initial disturbances.

Contribution

It establishes the time-asymptotic stability of a composite wave involving degenerate Oleinik shock and rarefaction for a non-convex flux system with Cattaneo's law, using novel energy estimates.

Findings

Proved stability under small wave strength and initial perturbations.

Applied Oleinik entropy condition and weighted energy estimates.

Demonstrated stability for a non-convex flux with hyperbolic-heat flux coupling.

Abstract

This paper examines the large-time behavior of solutions to a one-dimensional conservation law featuring a non-convex flux and an artificial heat flux term regulated by Cattaneo's law, forming a 2$\times$2 system of hyperbolic equations. Under the conditions of small wave strength and sufficiently small initial perturbations, we demonstrate the time-asymptotic stability of a composite wave that combines a degenerate Oleinik shock and a rarefaction wave. The proof utilizes the Oleinik entropy condition, the a-contraction method with time-dependent shifts, and weighted energy estimates.