Systems of conservation laws of mixed type

A local theory of weak solutions of first-order nonlinear systems of conservation laws is presented. In the systems considered, two of the characteristic speeds become complex for some achieved values of the dependent variable. The transonic “small disturbance” equation is an example of this class of systems. Some familiar concepts from the purely hyperbolic case are extended to such systems of mixed type, including genuine nonlinearity, classification of shocks into distinct fields and entropy inequalities. However, the associated entropy functions are not everywhere locally convex, shock and characteristic speeds are not bounded in the usual sense, and closed loops and disjoint segments are possible in the set of states which can be connected to a given state by a shock. With various assumptions, we show (1) that the state on one side of a shock plus the shock speed determine the state on the other side uniquely, as in the hyperbolic case; (2) that the “small disturbance” equation is a local model for a class of such systems; and (3) that entropy inequalities and/or the existence of viscous profiles can still be used to select the “physically relevant” weak solution of such a system.