Hyperbolic conservation laws and spacetimes with limited regularity

Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows andgeneral relativity. Recentworkby the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either be fixed (shallow water equations on the sphere, for instance) or be one of the unknowns of the theory (EinsteinEuler equations of general relativity). This work is especially concerned with solutions and manifolds with limited regularity. We review here results on three themes: (1) Shock wave theory for hyperbolic conservation laws on manifolds, developed jointly with M. Ben-Artzi (Jerusalem); (2) Existence of matter Gowdy-type spacetimes with bounded variation, developed jointlywith J. Stewart (Cambridge). (3) Injectivity radius estimates for Lorentzian manifolds under curvature bounds, developed jointly with B.-L. Chen (Guang-Zhou).