A Useful Class of Two-dimensional Conservation Laws 1. Conservation Laws with a Single Nondegenerate Wave Cone

We identify a class of two-dimensional systems, which includes, for example, standard compressible ow, shallow water equations and nonlinear wave equations, for which a quasi-steady similarity reduction leads to change of type similar to steady transonic ow. For such systems, we outline a strategy for solving Riemann problems. In studying two-dimensional Riemann problems, we have noted a structure which makes them amenable to study by a self-similar reduction. Consider a system of conservation laws in three independent variables (two space and one time), @ t H(U) + @ x F(U) + @ y G(U) = 0 ; (1) with U = (u 1 ; u 2 ; : : :; u n); H is usually the identity mapping or a projection. The quasilinear form of (1) is P(@; U)U (J@ t + A@ x + B@ y)U = 0 (2) where J(U) = dH(U), A(U) = dF(U) and B(U) = dG(U) are n n matrices depending on U. Suppose the system is hyperbolic; that is, the linearized symbol of P, p(; U) det jP(; U)j (3) (= (; ;) a covector in a space dual to (t; x; y) = ~ x) has hyperbolic or timelike directions ~ N for which p(~ N; U) > 0, while p(~ N +; U) = 0 has n real roots i () for each 2 IR n , 6 2 spanf ~ Ng, and the symbol matrix has a complete eigenstructure. The solutions of p(; U) = 0 form the normal cone. The class we study is described as follows. Assumption 1: We can write p as a product of n ? 2 linear (degenerate) factors and a single quadratic (nondegenerate) polynomial factor. The equations of isentropic and polytropic gas ow, as well as the two-dimensional shallow water equations, the unsteady transonic small disturbance equation (also called the two-dimensional Burgers equation), nonlinear wave equations, and unsteady full potential ow all have this form. E x a m p l e 1. The compressible Euler equations for a polytropic gas are @ t 0 B B @ u v E 1 C C A + @ x 0 B B @ u u 2 + p uv uE + up 1 C C A + @ y 0 B B @ v uv v 2 + p vE + vp 1 C C A = 0 ; where E = (u 2 + v …