The initial formation and structure of two-dimensional diffusive shock waves (Q1078413)

The aim of the author is to describe the formation of two-dimensional shock waves for quasilinear partial differential equations with weak dissipation. To this end he considers an equation of the form \[ u_ t+c_ 1(u)u_ x+c_ 2(u)u_ y=\epsilon d(u_{xx}+u_{yy}) \] subject to the following initial condition \(u(x,y,0)=f(x,y)\). He first shows that the characteristics and their singularity where the derivatives of the solution become infinite are described by a generic cubic equation. Then he derives some properties of the singularities.