Synchronisation for scalar conservation laws via Dirichlet boundary

TL;DR

This paper presents a new elementary proof of geometric synchronization for scalar conservation laws with Dirichlet boundary conditions, avoiding the strict maximum principle and using boundary dissipation estimates.

Contribution

It introduces a novel proof technique that does not rely on the maximum principle and identifies conditions for uniform estimates using super- and sub-solutions.

Findings

Synchronization occurs under certain coercivity conditions.

Boundary dissipation estimates are key to the proof.

The approach extends to cases without coercivity using energy estimates.

Abstract

We provide an elementary proof of geometric synchronisation for scalar conservation laws on a domain with Dirichlet boundary conditions. Unlike previous results, our proof does not rely on a strict maximum principle, and builds instead on a quantitative estimate of the dissipation at the boundary. We identify a coercivity condition under which the estimates are uniform over all initial conditions, via the construction of suitable super- and sub-solutions. In lack of such coercivity our results build on Lp energy estimates and a Lyapunov structure.