Traces for Functions of Bounded Variation on Manifolds with Applications to Conservation Laws on Manifolds with Boundary

Abstract

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 ∞ , 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.

Rigorous estimates on balance laws in bounded domains

“…Refer to [19,Section 2.6] and [20] for a generalization to L ∞ boundary data. For a definition of functions of bounded variation on a manifold, refer for instance to [14,Definition 3.1].…”

Weakly Coupled Systems of Conservation Laws on Moving Surfaces

“…Another analytically interesting problem is to consider surfaces with boundary and adequate boundary conditions in the context of our work. For the scalar problem on time independent manifolds there has been some work recently by Kröner, Müller and Strehlau in [20].…”

Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

“…Although investigations concerning well-posedness of evolution equations on manifolds attracted a significant amount of attention recently, this problem for degenerate parabolic equations on manifolds has not been considered until now. The most closely related research is directed towards scalar conservation laws on manifolds and we mention [4,20,25] for the Cauchy problem corresponding to scalar conservation laws on manifolds, and [17,26] for the (initial)-boundary value problem on manifolds. The approach in [26] is based on the kinetic formulation as well, and Definition 3.1. from there inspired our kinetic solution concept.…”