Scalar conservation laws on constant and time-dependent Riemannian manifolds

TL;DR

This paper proves well-posedness and key properties of scalar conservation laws on Riemannian manifolds with static or dynamic metrics, including existence, uniqueness, and total variation estimates for solutions.

Contribution

It extends the theory of scalar conservation laws to Riemannian manifolds with time-dependent metrics, allowing flux functions with divergence and providing new geometric criteria for total variation diminishing solutions.

Findings

Existence and uniqueness of entropy solutions on manifolds.

L^1 contraction and comparison principles established.

Total variation estimates derived for BV initial data.

Abstract

In this paper we establish well-posedness for scalar conservation laws on closed manifolds M endowed with a constant or a time-dependent Riemannian metric for initial values in L^\infty(M). In particular we show the existence and uniqueness of entropy solutions as well as the L^1 contraction property and a comparison principle for these solutions. Throughout the paper the flux function is allowed to depend on time and to have non-vanishing divergence. Furthermore, we derive estimates of the total variation of the solution for initial values in BV(M), and we give, in the case of a time-independent metric, a simple geometric characterisation of flux functions that give rise to total variation diminishing estimates.