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δ ' -shock waves as a new type of solutions to systems of conservation laws. (English) Zbl 1108.35116

Summary: A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called δ (n) -shock wave, where δ (n) is nth derivative of the Dirac delta function (n=1,2,). In this paper the case n=1 is studied in details. We introduce a definition of δ ' -shock wave type solution for the system

u t +f ( u ) x =0,v t +f ' (u) v x =0,w t +f '' (u) v 2 + 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. Finally, a solitary δ ' -shock wave type solution to the Cauchy problem of the system of conservation laws u t +(u 2 ) x =0, v t +2(uv) x =0, w t +2(v 2 +uw) x =0 with piecewise continuous initial data is constructed. These results first show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.

MSC:
35L67Shocks and singularities
35L65Conservation laws
76L05Shock waves; blast waves (fluid mechanics)