Long time behaviour of viscous scalar conservation laws

TL;DR

This paper investigates the long-term stability and behavior of stationary solutions in viscous scalar conservation laws with periodic flux, establishing stability results for shocks in unbounded domains and for periodic solutions.

Contribution

It proves the existence and stability of viscous stationary shocks in unbounded domains and confirms the stability of periodic stationary solutions in bounded periodic domains.

Findings

Existence of viscous stationary shocks connecting different solutions.

Stability of shocks in L^1 under certain conditions.

Stability of periodic solutions with uniform bounds.

Abstract

This paper is concerned with the stability of stationary solutions of the conservation law $\partial_t u + \mathrm{div}_y A(y,u) -\Delta_y u=0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviours are studied here: the case when the equation is set on $\R$, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of viscous stationary shocks - also called standing shocks - which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable in $L^1$, provided the initial disturbance satisfies some appropriate boundedness conditions. We also extend this result to arbitrary initial data, but with some restrictions on the flux $A$. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniform $L^\infty$ bounds on the solution of the conservation law, and on sub- and super-solution techniques.