Long-time behavior in scalar conservation laws

TL;DR

This paper studies the long-term behavior of solutions to scalar conservation laws on Riemannian manifolds, showing convergence to average values on a torus under certain conditions.

Contribution

It demonstrates convergence of entropy solutions to their average on a torus with weak non-linearity assumptions, extending understanding of long-time dynamics.

Findings

Solutions converge to their average value in L^p on the torus.

Weak non-linearity of flux suffices for convergence.

Partial results are provided for general manifolds.

Abstract

We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine non-linearity of the flux, the solution converges to its average value in $L^{p}$, $1\leq p<+\infty$. We give a partial result in the general case.