Stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for a non-convex viscoelastic system

Abstract

We investigate the time-asymptotic stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for non-convex viscoelastic system. By choosing delicate time-dependent shift functions to the viscous shock and suitable weight functions for the relative entropy, we succeed in proving the time-asymptotic stability of the composite of a viscous Liu shock and a rarefaction wave from two different characteristic families for the one-dimensional viscoelastic system with a cubic non-convex stress-strain relation by $a$-contraction method. Here the non-classical Liu shock can be either degenerate or non-degenerate, which can be seen as the system version of the scalar Oleinik shock. Note that both the shift and the weight functions are chosen quite differently in the two cases of nearly degenerate and fully non-degenerate. The paper represents the first time-asymptotic stability result for the composite waves consisting of both non-classical Liu shock and rarefaction wave from two different families for the non-convex physical system.