Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds

  • Matania Ben-Artzi

    Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
  • Philippe G. LeFloch

    Laboratoire Jacques-Louis Lions & CNRS UMR 7598, University of Paris VI, 75252 Paris, France


Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field 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 measure-valued mappings. We establish the well-posedness 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 geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.

Cite this article

Matania Ben-Artzi, Philippe G. LeFloch, Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 6, pp. 989–1008

DOI 10.1016/J.ANIHPC.2006.10.004