Resonances for a Contact Wave

Systems of three or more conservation laws in which a conserved quantity is passively transported have a contact eld; sometimes, the contact speed and the speed of one of the nonlinear waves coincide on a surface in state space. We prove that this resonance surface contains the equilibrium points of the contact eld. We construct locally the Riemann solution for an example of such a system of three equations, which arises in ow in petroleum reservoirs. Solutions resulting from the resonating waves give rise to new objects, which we call wave surfaces. The complete Riemann solution is obtained by putting together transversely the wave surface and the remaining non resonant waves. In this example, the contact eld has saddle points, and a new wave appears, a transitional contact wave. In the presence of this new wave, the Riemann solution for the system of three equations possesses four rather than three wave groups.