Scalar conservation laws on moving hypersurfaces

TL;DR

This paper establishes existence and uniqueness of entropy solutions for scalar conservation laws on moving hypersurfaces, introduces a regularization approach, and explores shock phenomena including geometrically induced shocks.

Contribution

It extends scalar conservation law theory to moving hypersurfaces, defining entropy solutions and proving key properties using parabolic regularization.

Findings

Existence and uniqueness of solutions proven.

Introduction of entropy solutions for moving hypersurfaces.

Numerical experiments demonstrating shock formation.

Abstract

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.