Singular solutions to systems of conservation laws: shocks, δ- and δ′-shocks

Using the definitions of δand δ′-shocks for the systems of conservation laws [12], [13], [39], the Rankine–Hugoniot conditions for δand δ′-shocks are derived. We present a construction of solutions to the Cauchy problems admitting δand δ′-shocks. In particular, the Riemann problem admitting shocks, δ-shocks, δ′-shocks, and vacuum states is considered. The geometric aspects of δand δ′-shocks are studied. Balance relations connected with area transportation, in particular, mass and momentum transportation relations for the zero-pressure gas dynamics system, are derived. We also study the algebraic aspects of δand δ′-shocks. Namely, the flux-functions of δand δ′-shock solutions are computed. Though the flux-functions are nonlinear, they can be considered as “right” singular superpositions of distributions thus being well defined Schwartzian distributions. Therefore, singular solutions of the Cauchy problems generate algebraic relations between distributional components of these singular solutions. The validity and naturalness of the above-mentioned definitions of δand δ′-shocks are discussed.