Time-periodic boundary effects on the shocks for scalar conservation laws

TL;DR

This paper investigates how time-periodic boundary conditions influence the long-term behavior of shocks in scalar conservation laws, revealing the formation of a propagating boundary wave that affects shock dynamics.

Contribution

It provides a rigorous analysis of the impact of time-periodic boundary conditions on shock stability and long-term behavior in scalar conservation laws, a previously unclear area.

Findings

Asymptotic states are governed by shifted shocks and boundary-induced periodic solutions.

Identification of a propagating boundary wave affecting shock dynamics.

Extension of shock stability analysis to time-periodic boundary conditions.

Abstract

This paper is concerned with the asymptotic stabilities of the inviscid and viscous shocks for the scalar conservation laws on the half-line $(-\infty,0)$ with shock speed $s<0$, subjected to the time-periodic boundary condition, which arises from the classical piston problems for fluid mechanics. Despite the importance, how time-periodic boundary conditions affect the long-time behaviors of Riemann solutions has remained unclear. This work addresses this gap by rigorously proving that in both inviscid and viscous case, the asymptotic states of the solutions under the time-periodic boundary conditions are not only governed by the shifted background (viscous) shocks, but also coupled with the time-periodic boundary solution induced by the time-periodic boundary. Our analysis reveals that these effects manifest as a propagating "boundary wave", which influences the shock dynamics.