Abstract: A concept of a new type of singular solutions to hyperbolic systems of conservation laws is introduced. It is so-called δn-shock wave, where δn is n-th derivative of the delta function.
We introduce a definition of δ'-shock wave type solution for the systemut+ f(u)x=0, vt+(f'(u)v)x=0, wt+(f''(u)v2+f'(u)w)x=0.Within the framework of this definition, the Rankine--Hugoniot conditions for δ'-shock are derived and analyzed from geometrical point of view. We prove δ'-shock balance relations connected with area transportation. A solitary δ'-shock wave type solution to the Cauchy problem of the system of conservation lawsut+(u2)x=0, vt+2(uv)x=0, wt+2(v2+uw)x=0with piecewise continuous initial data is constructed.
These results show that solutions of hyperbolic systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.
Conservation Laws Preprint Server <email@example.com> 2005-02-15 13:36:49 UTC