$\delta$- and $\delta'$-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes

This is a survey of some results and problems connected with the theory of generalized solutions of quasi-linear conservation law systems which can admit delta-shaped singularities. They are the so-called $\delta$-shock wave type solutions and the recently introduced $\delta^{(n)}$-shock wave type solutions, $n=1,2,\dots$, which cannot be included in the classical Lax–Glimm theory. The case of $\delta$- and $\delta'$-shock waves is analyzed in detail. A specific analytical technique is developed to deal with such solutions. In order to define them, some special integral identities are introduced which extend the concept of weak solution, and the Rankine–Hugoniot conditions are derived. Solutions of Cauchy problems are constructed for some typical systems of conservation laws. Also investigated are multidimensional systems of conservation laws (in particular, zero-pressure gas dynamics systems) which admit $\delta$-shock wave type solutions. A geometric aspect of such solutions is considered: they are connected with transport and concentration processes, and the balance laws of transport of ‘volume’ and ‘area’ to $\delta$- and $\delta'$-shock fronts are derived for them. For a ‘zero-pressure gas dynamics’ system these laws are the mass and momentum transport laws. An algebraic aspect of these solutions is also considered: flux-functions are constructed for them which, being non-linear, are nevertheless uniquely defined Schwartz distributions. Thus, a singular solution of the Cauchy problem generates algebraic relations between its components (distributions).