Time-periodic boundary effects on the shocks for scalar conservation laws
Yuan Yuan
TL;DR
This work analyzes the impact of time-periodic boundary conditions on shock dynamics for scalar conservation laws on the half-line. It reveals a boundary-wave mechanism: in the inviscid case, the solution converges to a shifted background shock plus a persisting time-periodic boundary wave, with the shock position $X(t)$ satisfying $|X(t)-st-X_\infty|\le C/\sqrt{t}$; in the viscous case, the solution converges to the superposition of a shifted viscous shock and the boundary wave, via an anti-derivative formulation and a carefully constructed ansatz. The analysis combines generalized characteristics, time-periodic solution theory, and weighted energy methods to establish precise asymptotics and robust stability results. The findings illuminate how boundary-periodic forcing can propagate into the bulk as a boundary wave, affecting shock dynamics and inducing a nontrivial long-time displacement.
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.