Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres

Abstract

" We study the singular structure of a kind of three dimensional non-selfsimilar global solutions and their interaction for quasilinear hyperbolic conservation laws. The initial discontinuity is two disjoint unit spheres and initial data just contain two different constant states, the global solutions and some new phenomena are discovered. We give the solutions in 0 \u2039 t \u2039 T^* and T^* \u2039 t, and at t=T^*, the two basic shock waves and the constant state u_ are disappeared.Then, we find a new shock wave between two rarefaction by R-H condition. Finally, we show the limit of the solution when t \u2192 \u221e. A technique is proposed to construct the three dimensional shock wave without dimensional reduction or coordinate transformation."