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

Yu. G. Rykov

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 1, page 55-78
  • ISSN: 0392-4041

Abstract

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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.

How to cite

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Rykov, Yu. G.. "On the nonhamiltonian character of shocks in 2-D pressureless gas." Bollettino dell'Unione Matematica Italiana 5-B.1 (2002): 55-78. <http://eudml.org/doc/194672>.

@article{Rykov2002,
abstract = {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.},
author = {Rykov, Yu. G.},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {generalized Rankin-Hugoniot conditions; shock-wave singularities},
language = {eng},
month = {2},
number = {1},
pages = {55-78},
publisher = {Unione Matematica Italiana},
title = {On the nonhamiltonian character of shocks in 2-D pressureless gas},
url = {http://eudml.org/doc/194672},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Rykov, Yu. G.
TI - On the nonhamiltonian character of shocks in 2-D pressureless gas
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/2//
PB - Unione Matematica Italiana
VL - 5-B
IS - 1
SP - 55
EP - 78
AB - 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.
LA - eng
KW - generalized Rankin-Hugoniot conditions; shock-wave singularities
UR - http://eudml.org/doc/194672
ER -

References

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