In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure. This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of twodimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u+ > 0, while explicit non-selfsimilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u+ < 0. The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension.
We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of Ck+1 of first category, defined in the range of the initial datum.
Two-dimensional Riemann problem for scalar conservation law are investigated and classification of global structure for its non-selfsimilar solution is given by analysis of structure and classification of envelope for non-selfsimilar 2D rarefaction wave. Initial data has two different constant states which are separated by initial discontinuity. We propose the concepts of plus envelope, minus envelope and mixed envelope, and some new structures and evolution phenomena are discovered by use of these concepts.
