Uniqueness for Multidimensional Hyperbolic Systems with Commuting Jacobians

We consider nonlinear hyperbolic systems of conservation laws in several space dimensions with Jacobian matrices that commute, and more generally systems that need not be conservative. Generalizing a theorem by Bressan and LeFloch for one-dimensional systems, we establish that the Cauchy problem admits at most one entropy solution depending continuously upon its initial data. The uniqueness result is proven within the class (introduced here) of locally regular BV functions with locally controlled oscillation. These regularity conditions are modeled on well-known properties in the one-dimensional case. Our uniqueness theorem also improves upon the known results for one-dimensional systems.