Structures of Interaction of Non-selfsimilar Elementary Waves for 2D Scalar Conservation Law with Two Initial Discontinuities

Abstract

In this paper, we investigate the global solution and the structures of interaction between two dimensional non-selfsimilar shock wave and rarefaction wave of general two-dimensional scalar conservation law in which flux functions f(u) and g(u) do not need to be convex, and the initial value contains three constant states which are respectively separated by two general initial discontinuities. When initial value contains three constant states, the cases of selfsimilar shock wave and rarefaction wave had been studied before, but no results of the cases of neither non-selfsimilar shock wave or non-selfsimilar rarefaction wave. Under the assumption that Condition H which is generalization of one dimensional convex condition, and some weak conditions of initial discontinuity, according to all the kinds of combination of elementary waves respectively staring from two initial discontinuities, we get four cases of wave interactions as S + S, S + R, R + S and R + R. By studying these interactions between non-selfsimilar elementary waves, we obtain and prove all structures of non-selfsimilar global solutions for all cases.