Scalar conservation laws on constant and time-dependent Riemannian manifolds

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.