UNIQUENESS VIA THE ADJOINT PROBLEMS FOR SYSTEMS OF CONSERVATION-LAWS

We prove a result of uniqueness of the entropy weak solution to the Ca uchy problem for a class of nonlinear hyperbolic systems of conservati on laws that includes in particular the p-system of isentropic gas dyn amics. Our result concerns weak solutions satisfying the, as we call i t, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock wav es and rarefaction waves present in a solution. For the proof of uniqu eness, we derive an existence result (respectively a uniqueness result ) for the backward (respectively forward) adjoint problem associated w ith the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with disco ntinuous coefficients. (C) 1993 John Wiley and Sons, Inc.