On delta-shocks and singular shocks

It is well known that there are “nonclassical” situations where, in contrast to Lax’s and Glimm’s results,systems of conservation laws may admit singular solutions (δ-shocks and singular shocks) such that their components contain delta functions [ASh05], [B94], [DSh03]– [LW02], [S02]– [Sh04], [TZZ94]. The exact structure of such type solutions is given below in (2), (7) and Definition 1. The theory of δ-shocks and singular shocks has been intensively developed in the last ten years. Moreover, in the recent papers [PSh06], [Sh06] the theory of δ′-shocks was established, and a concept of δ-shocks was introduced, n = 2, 3, . . .. They are new type singular solutions such that their components contain delta functions and their derivatives. In the δ-shock (singular shock) and δ′-shock theories there are many open and complicated problems. One of them is connected with the concept of singular shocks. Some problems related with singular shocks were studied in [K99]– [KK90], [S02], [S03]. A model system admitting a singular shock is the well-known Keyfitz-Kranzer system