On Delta-Shocks and Singular Shocks

V. M. Shelkovich ()
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V. M. Shelkovich: St. Petersburg State Architecture and Civil Engineering University, Department of Mathematics

A chapter in Hyperbolic Problems: Theory, Numerics, Applications, 2008, pp 971-979 from Springer

Abstract: It is well known that there are “nonclassical” situations where, in contrast to Lax’s and Glimm’s results, the Cauchy problem for a system of conservation laws does not possess a weak L ∞-solution except for some particular initial data. To solve the Cauchy problem in this “nonclassical” situation, it is necessary to introduce new singular solutions called δ-shocks and singular shocks. The components of these solutions 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 10 years. In particular, in numerous papers δ-shock type solutions of “zero-pressure gas dynamics” have been studied. Moreover, in the recent papers [PSh06], [Sh06] the theory of δ′-shocks was established, and a concept of δ(n)-shocks was introduced, n = 2, 3,. … They are new type singular solutions such that their components contain delta functions and their derivatives.

Date: 2008
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DOI: 10.1007/978-3-540-75712-2_102

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