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

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