2001/05/15 by Yuri G. Rykov
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.MP #msc:35L65 #msc:35L67 #msc:46F10
published as Bolletino dell' U.M.I., Sezione B, (8), 5-B, 2002, 55-78 · 25 pages, 1 figure, to be published in Bolletino Unione Matematica Italiana
arxiv created 2001/05/15 · arxiv updated 2009/11/30
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.