Nonlinear Resonance in Systems of Conservation Laws

The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed. The inhomogeneity is modeled by a linearly degenerate field. The solution of the Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinear systems of conservation laws. Generic conditions on the fluxes are stated and it is proved that under these conditions, the solution of the Riemann problem exists, is unique, and has a fixed structure; this demonstrates that, in the above sense, resonant inhomogeneous systems generically have the same canonical form. The wave curves for these systems are only Lipschitz continuous in a neighborhood of the states where the wave speeds coincide, and so, in contrast to strictly hyperbolic systems, the implicit function theorem cannot be applied directly to obtain existence and uniqueness. Here we show that existence and uniqueness for the Riemann problem is a consequence of the uniqueness of intersection points of Lipschitz continuous manifolds of complementary dimensions. These systems are resonant for two reasons: The linearized problem exhibits classical resonant behavior, while the nonlinear initial value problem exhibits a “nonlinear resonance” in the sense that wave speeds from different families of waves are not distinct; so the number of times a pair of waves can interact in a given solution cannot be bounded a priori. Since waves are reflected in other families every time a pair of waves interact, a proliferation of reflected waves can occur by the interaction of a single pair of waves. Examples of resonant inhomogeneous systems are observed in model problems for the flow of a gas in a variable area duct and in Buckley–Leverett systems that model multiphase flow in a porous medium.