Solutions to quasilinear hyperbolic conservation laws with initial discontinuities

Abstract

We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t <T*. Then, when T*≤ t <T', we get a new shock wave between two rarefactions, and then, when t>T', another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t→∞. The technique does not involve dimensional reduction or coordinate transformation.