Traces for functions of bounded variation on manifolds with applications to conservation laws on manifolds with boundary

TL;DR

This paper establishes the existence of traces for BV functions on Riemannian manifolds with boundary, enabling analysis of conservation laws with boundary conditions using BV techniques and vanishing viscosity.

Contribution

It introduces a trace existence result for BV functions on manifolds with boundary and applies it to prove well-posedness of conservation laws with boundary conditions.

Findings

Existence of bounded trace for BV functions on manifolds with boundary

Well-posedness of scalar conservation laws on manifolds with boundary

Total variation estimates for solutions

Abstract

In this paper we show existence of a trace for functions of bounded variation on Riemannian manifolds with boundary. The trace, which is bounded in $L^\infty$, is reached via $L^1$-convergence and allows an integration by parts formula. We apply these results in order to show well-posedness and total variation estimates for the initial boundary value problem for a scalar conservation law on compact Riemannian manifolds with boundary in the context of functions of bounded variation via the vanishing viscosity method. The flux function is assumed to be time-dependent and divergence-free.