Formula omitted]-Shock waves as a new type of solutions to systems of conservation laws

Abstract

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 ( u v ) x = 0 , w t + 2 ( v 2 + u w ) 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.