Wellposedness theory for geometry compatible hyperbolic conservation laws on manifolds
Abstract
Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to nonlinear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vectorfield on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measurevalued mappings. We establish the wellposedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of {\sl geometrycompatible} (as we call it) conservation laws is singled out as an important case of interest, which leads to robust $L^p$ estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the $L^1$ contraction property and leads to a unique contractive semigroup of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2006
 arXiv:
 arXiv:math/0612846
 Bibcode:
 2006math.....12846B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Numerical Analysis;
 35L65;
 58J45;
 76N10
 EPrint:
 30 pages. This is Part 1 of a series