On the nonhamiltonian character of shocks in 2-D pressureless gas

Details On the nonhamiltonian character of shocks in 2-D pressureless gas On the nonhamiltonian character of shocks in 2-D pressureless gas

2001-05-15

arxiv.org/abs/math/0105118v1

Bolletino dell' U.M.I., Sezione B, (8), 5-B, 2002, 55-78

Mathematics

Analysis of PDEs

25 pages, 1 figure, to be published in Bolletino Unione Matematica Italiana

Scientific paper

The paper deals with the 2-D system of gas dynamics without pressure which was introduced in 1970 by Ya.Zeldovich to describe the formation of large-scale structure of the Universe. Such system occurs to be an intermediate object between the systems of ordinary differential equations and hyperbolic systems of PDE. The main its feature is the arising of singularities: discontinuities for velocity and delta functions of various types for density. The rigorous notion of generalized solutions in terms of Radon measures is introduced and the generalization of Rankine-Hugoniot conditions is obtained. On the basis of such conditions it is shown that the variational representation for the generalized solutions, which is valid for 1-D case, in 2-D case generally speaking does not take place. A nontrivial 1-D system of nonstrictly hyperbolic type is also obtained to describe the evolution inside the shock.

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