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

In this paper, we show the existence of a trace for functions of bounded variation on Riemannian manifolds with boundary. The trace, which is bounded in L^∞, is reached via L¹- 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 timedependent and divergence-free.