Conservation Laws in Two Space Dimensions

Abstract

As noted earlier in this book, the general theory of nonlinear hyperbolic systems of conservation laws in several space dimensions is terra incognita. Nevertheless, a number of important problems in two space dimensions are currently tractable, as they admit stationary or self-similar solutions, in which case the number of independent variables is reduced to two. The chapter begins with an introduction to the Riemann problem for scalar conservation laws in two space dimensions. The resulting equation in the (two) selfsimilar variables retains hyperbolicity. The emerging wave pattern is quite intricate and, depending on the data, may assume any one of 32 distinct configurations, of which two representative cases will be recorded here. The next task is to consider stationary or self-similar solutions of the Euler equations for planar isentropic gas flow. The number of independent variables is again reduced to two; however, the price to pay is that the resulting system is no longer hyperbolic but ofmixed elliptic-hyperbolic type. In a transonic flow, the border between the elliptic and the hyperbolic region is a free boundary that has to be determined as part of the solution. The recently solved, classical problem of regular shock reflection by a wedge, for irrotational flow, will be discussed in some detail, as an illustrative example of this type. Additional examples of systems of conservation laws of mixed elliptic-hyperbolic type, arising in fluid dynamics or differential geometry, are found in the literature cited in Section 17.7.