Related Problems for TV-estimates for Conservation Laws on Surfaces

摘要

The study of nonlinear conservation laws on surfaces is motivated by many applications such as geophysical flows, general relativity, fractions in porous media, sound waves on surfaces and transport processes on cells. Since huge difficulties in the theoretical analysis of systems of nonlinear conservation laws arise, even in the Euclidean space, we focus on nonlinear scalar conservation laws posed on closed Riemannian manifolds. The fluxes of the conservation laws are modelled by vector fields on the manifold which depend both on the unknown function as a parameter and on the space variable, since the vector fields have to be in the tangent space of the manifold at every point. In order to prove error estimates for finite volume schemes, one requires estimates of the total variation of the solution. In this contribution we present a proof (that is different from the one in [4]) for an estimate of the total variation for the two dimensional case. Therefore we consider the viscous approximation of the conservation law. Due to the space-dependence of the Laplace- Beltrami operator, new second-order derivatives in the unkown occur. The key idea to control these terms is to reuse the differential equation in a sophisticated way. The result is an estimate of the total variation of the entropy solution of the conservation law. We construct an example which shows that we cannot expect a TVD property on a manifold. This is solely a consequence of the fact that the flux-vector depends on the spatial coordinate.n addition we implemented the finite volume scheme proposed by Amorim et al. [1] for the case of a sphere. On the basis of Ben-Artzi et al. [3] we constructed a class of test problems which are equivalent to one-dimensional conservation laws in order to calculate EOCs.