Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws

TL;DR

This paper proves the time-asymptotic stability of a composite wave consisting of a degenerate Oleinik shock and a rarefaction wave for 1D cubic non-convex viscous conservation laws, introducing a novel $a$-contraction method with a time-dependent shift.

Contribution

It develops a new $a$-contraction method with a suitable weight function and a time-dependent shift to prove stability of complex wave interactions in non-convex conservation laws.

Findings

Established stability of composite wave with large shock strength.

Introduced a new $a$-contraction method for wave stability.

Demonstrated sub-linear growth of the shift function over time.

Abstract

We are concerned with the large-time behavior of the solution to one-dimensional (1D) cubic non-convex scalar viscous conservation laws. Due to the inflection point of the cubic non-convex flux, the solution to the corresponding inviscid Riemann problem can be the composite wave of a degenerate Oleinik shock and a rarefaction wave and these two nonlinear waves are always attached together. We give a first proof of the time-asymptotic stability of this composite wave, up to a time-dependent shift to the viscous Oleinik shock, for the viscous equation. The Oleinik shock wave strength can be arbitrarily large. The main difficulty is due to the incompatibility of the time-asymptotic stability proof framework of individual viscous shock by the so-called anti-derivative method and the direct $L^2$-energy method to rarefaction wave. Here we develop a new type of $a$-contraction method with suitable weight function and the time-dependent shift to the viscous shock, which is motivated by [9,12]. Another difficulty comes from that the Oleinik shock and rarefaction wave are always attached together and their wave interactions are very subtle. Therefore, the same time-dependent shift needs to be equipped to both Oleinik shock and rarefaction wave such that the wave interactions can be treated in our stability proof. Time-asymptotically, this shift function grows strictly sub-linear with respect to the time and then the shifted rarefaction wave is equivalent to the original self-similar rarefaction wave.