Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems

In this paper we establish a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one-space dimension. First of all we introduce a new notion of arl,admissible solutions based on prescribed sets of admissible discontinuities Phi and admissible speeds psi. Our definition unifies in a single framework the Various notions of entropy solutions known for hyperbolic systems of conservation laws, as well as for systems in nonconservative form. For instance, it covers the nonclassical (undercompressive) shock waves generated by a vanishing diffusion-dispersion regularization and characterized via a kinetic relation. It also covers Dal Maso, LeFloch, and Murat's definition of weak solutions of nonconservative systems. Under certain natural assumptions on the prescribed sets Phi and psi and assuming the existence of a L-1-continuous semi-group of admissible solutions, we prove that, for each Cauchy datum at t=0, there exists at most one admissible solution to the Cauchy problem depending L-1-continuously upon the initial data. In particular, our result shows the uniquness of the L-1-continuous semi-group of admissible solutions. In short, this paper proves that supplementing a hyperbolic system H ith the "dynamics" of elementary discontinuities characterizes at most one L-1-continuaus and admissible solution. (C) 2001 Academic Press.