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Applicable Analysis
An International Journal
Volume 90, 2011 - Issue 5
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Original Articles

Mass, momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

, &
Pages 831-842 | Received 07 May 2010, Accepted 18 Aug 2010, Published online: 25 Jan 2011

References

  • Korchinski , DJ . 1977 . Solution of a Riemann problem for 2 × 2 systems of conservation laws possessing no classical weak solution, Ph.D. Thesis , Garden City, NY : Adelphi University .
  • Zeldovich , YB and Myshkis , AD . Elements of Mathematical Physics. Medium Consisting of Non-interacting Particles, Nauka, moscow, 1973 (In Russian)
  • Kraiko , AN . 1979 . Discontinuity surfaces in medium without self-pressure . Prikl. Math. Mech. , 43 : 539 – 449 . (In Russian)
  • Albeverio , S and Shelkovich , VM . 2005 . “ On the delta-shock front problem ” . In Analytical Approaches to Multidimensional Balance Laws, Ch. 2 , Edited by: Rozanova , OS . 45 – 88 . New York : Nova Science Publishers, Inc. .
  • Bouchut , F . 1994 . “ On zero pressure gas dynamics ” . In Advances in Kinetic Theory and Computing , Edited by: Perthame , B . Vol. 22 , 171 – 190 . Singapore : Series on Advances in Mathematics for Applied Sciences, World Scientific .
  • Chen , GQ and Liu , H . 2003 . Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids . SIAM J. Math. Anal. , 34 : 925 – 938 .
  • Chen , GQ and Liu , H . 2004 . Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids . Physica D , 189 : 141 – 165 .
  • Danilov , VG and Shelkovich , VM . 2005 . Delta-shock wave type solution of hyperbolic systems of conservation laws . Quart. Appl. Math. , 63 ( 3 ) : 401 – 427 .
  • Danilov , VG and Shelkovich , VM . 2005 . Dynamics of propagation and interaction of delta-shock waves in conservation law systems . J. Differ. Equ. , 211 : 333 – 381 .
  • Weinan , E , Rykov , Yu and Sinai , YG . 1996 . Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics . Comm. Math. Phys. , 177 : 349 – 380 .
  • Li , J and Zhang , T . 1998 . On the initial-value problem for zero-pressure gas dynamics, Hyperbolic problems: Theory, Numerics, Applications . Seventh International Conference in Zürich . February 1998 . pp. 629 – 640 . Basel, Boston, Berlin : Birkhäuser Verlag .
  • Nedeljkov , M . 2010 . Shadow waves: Entropies and interactions for delta and singular shocks . Arch. Ration. Mech. Anal. , 197 : 489 – 537 .
  • Nedeljkov , M and Oberguggenberger , M . 2006 . Interactions of delta shock waves in a strictly hyperbolic system of conservation laws . J. Math. Anal. Appl. , 344 ( 2 ) : 1143 – 1157 .
  • V. , M . Shelkovich, δ-and δ′-shock types of singular solutions to systems of conservation laws and the transport and concentration processes, Uspekhi Mat. Nauk, 63:3(381) (2008), 73–146 (English transl. in Russian Math. Surveys, 63(3) (2008), 473–546)
  • Shen , W and Zhang , T . 1999 . The Riemann problem for the transportation equations in gas dynamics . Mem. Amer. Math. Soc. , 137 ( 654 ) : 1 – 77 .
  • Shandarin , SF and Zeldovich , YaB . 1989 . The large-scale structure of the universe: urbulence, intermittence, structures in self-gravitating medium . Rev. Mod. Phys. , 61 : 185 – 220 .
  • Zeldovich , YaB . 1970 . Gravitational instability: An approximate theory for large density perturbations . Astron. Astrophys. , 5 : 84 – 89 .
  • Berthelin , F , Degond , P , Delitala , M and Rascle , M . 2008 . A model for the formation and evolution of traffic jams . Arch. Rat. Mech. Anal. , 187 ( 2 ) : 185 – 220 .
  • Edwards , CM , Howinson , SD , Ockendon , H and Ockendon , JR . 2008 . Non-classical shallow water flows . J. Appl. Math. , 73 : 137 – 157 .
  • Kraiko , AN and Sulaimanova , SM . 1983 . Two-phase flows of a gas-particle mixture near impermeable surfaces with the formation of ‘sheets’ and ‘filaments’ . Prikl. Math. Mekh. , 47 ( 4 ) : 619 – 630 . (In Russian)
  • Kraiko , AN . 1982 . On two-phase flows model of gas and dispersed in it particles . Prikl. Math. Mekh. , 46 ( 1 ) : 96 – 106 . (In Russian)
  • Osiptsov , AN . 1984 . Investigation of regions of unbounded growth of the particle concentration in dispersi flows . Fluid Dyn. , 19 ( 3 ) : 378 – 385 .
  • A. N. Osiptsov, Modified Lagrangian method for calculating the particle concentration in dusty-gas flows with intersecting particle trajectories, Proceedings of the 3d Internatinal Conference on Multiphase Flows, Lyon, France, CD-ROM ‘ICMF’ 98’, 1998, paper 236, p. 8.
  • Osiptsov , AN . 2000 . Lagrangian modeling of dust admixture in gas flows . Astrophys. Space Sci. , 274 : 377 – 386 .
  • Nilsson , B and Shelkovich , VM . 2010 . Mass, momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks . Appl. Anal. ,
  • Sever , M . 2001 . An existence theorem in the large for zero-pressure gas dynamics . Differ. integral equ. , 14 ( 9 ) : 1077 – 1092 .
  • Kanwal , RP . 1998 . Generalized Functions: Theory and technique , Berlin : Birkhäuser, Boston, Basel .
  • Albeverio , S , Korshunova , A and Rozanova , O . Probabilistic model associated with the pressureless gas dynamics . Preprint (eprint), Available at: arXiv:0908.2084
  • Shelkovich , VM . 2006 . The Rankine–Hugoniot conditions and balance laws for δ-shocks . Fundamentalnaya Prikl. Mate. , 12 ( 6 ) : 213 – 229 . (English transl. in: J. Math. Sci., Springer US, 151 (2008), pp. 2781–2792)

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