Far East Journal of Mathematical Sciences, vol.31, pp.49-83, 2008 (Refereed Journals of Other Institutions)
We consider nonlinear hyperbolic conservation laws, posed on a differential
(n + 1)-manifold with boundary referred to as a spacetime, and
in which the “flux” is defined as a flux field of n-forms depending on a
parameter (the unknown variable). We introduce a formulation of the initial
and boundary value problem which is geometric in nature and is more
natural than the vector field approach recently developed for Riemannian
manifolds. Our main assumption on the manifold and the flux field is a
global hyperbolicity condition, which provides a global time-orientation
as is standard in Lorentzian geometry and general relativity. Assuming
that the manifold admits a foliation by compact slices, we establish the
existence of a semi-group of entropy solutions. Moreover, given any two
hypersurfaces with one lying in the future of the other, we establish a
“contraction” property which compares two entropy solutions, in a (geometrically
natural) distance equivalent to the L1 distance. To carry out
the proofs, we rely on a new version of the finite volume method, which
only requires the knowledge of the given n-volume form structure on the
(n + 1)-manifold and involves the total flux across faces of the elements
of the triangulations, only, rather than the product of a numerical flux
times the measure of that face.