  We consider conservation laws on moving hypersurfaces. In this work the
velocity of the surface is prescribed. But one may think of the velocity to be
given by PDEs in the bulk phase. We prove existence and uniqueness for a scalar
conservation law on the moving surface. This is done via a parabolic
regularization of the hyperbolic PDE. We then prove suitable estimates for the
solution of the regularized PDE, that are independent of the regularization
parameter. We introduce the concept of an entropy solution for a scalar
conservation law on a moving hypersurface. We also present some numerical
experiments. As in the Euclidean case we expect discontinuous solutions, in
particular shocks. It turns out that in addition to the "Euclidean shocks"
geometrically induced shocks may appear.

Analysis of PDEs