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On Delta-Shocks and Singular Shocks

In: Hyperbolic Problems: Theory, Numerics, Applications

Author

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  • V. M. Shelkovich

    (St. Petersburg State Architecture and Civil Engineering University, Department of Mathematics)

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.

Suggested Citation

  • V. M. Shelkovich, 2008. "On Delta-Shocks and Singular Shocks," Springer Books, in: Sylvie Benzoni-Gavage & Denis Serre (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 971-979, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-75712-2_102
    DOI: 10.1007/978-3-540-75712-2_102
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