Uniqueness of Classical and Nonclassical Solutions for Nonlinear Hyperbolic Systems

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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 admissible solutions based on prescribed sets of admissible discontinuitiesΦ and admissible speedsψ. 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 Φ and ψ and assuming the existence of a L1-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 L1-continuously upon the initial data. In particular, our result shows the uniqueness of the L1-continuous semi-group of admissible solutions. In short, this paper proves that supplementing a hyperbolic system with the “dynamics” of elementary discontinuities characterizes at most one L1-continuous and admissible solution.