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

Rykov, Yu. G.

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 1, pp. 55-78.

Abstract (VO)

The paper deals with the 2-D system of gas dynamics without pressure which was introduced in 1970 by Ua. Zeldovich to describe the formation of largescale 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 d-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.

Riassunto

Si considera un sistema bidimensionale della dinamica dei gas introdotto nel 1970 da Ya. Zeldovich per descrivere la formazione della struttura di grande scala dell'universo. Il sistema si rivela come qualcosa di intermedio tra un sistema di equazioni differenziali ordinarie e un sistema iperbolico di equazioni alle derivate parziali. La caratteristica principale è la nascita di singolarità: discontinuità della velocità e funzioni delta di vario tipo per la densità. Si dà una descrizione rigorosa delle soluzioni generalizzate in termini di misure di Radon e si ottiene una generalizzazione delle condizione di Rankine-Hugoniot. Sulla base di tali condizioni si mostra che la rappresentazione variazionale delle soluzioni generalizzate, valida nel caso unidimensionale, non vale in generale nel caso bidimensionale. Si ottiene anche un sistema unidimensionale non banale non strettamente iperbolico per la descrizione dell'evoluzione all'interno dell'urto.

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