Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks

References

• Evans , LC . 1998 . Partial Differential Equations , Providence , Road Island : American Mathematical Society .
   Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Bouchut , F . 1994 . “ On zero pressure gas dynamics ” . In Series on Advances in Mathematics for Applied Sciences, ‘Advances in Kinetic Theory and Computing’ , Edited by: Perthame , B . Vol. 22 , 171 – 190 . Singapore : World Scientific .
   Google Scholar
• 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 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Danilov , VG and Shelkovich , VM . 2005 . Delta-shock wave type solution of hyperbolic systems of conservation laws . Quart. Appl. Math. , 63 ( 3 ) : 401 – 427 .
   Web of Science ®Google Scholar
• Danilov , VG and Shelkovich , VM . 2005 . Dynamics of propagation and interaction of delta-shock waves in conservation law systems . J. Diff. Eqns , 211 : 333 – 381 .
   Web of Science ®Google Scholar
• Weinan , E , Rykov , Yu and Sinai , YaG . 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 .
   Web of Science ®Google Scholar
• Albeverio , S and Shelkovich , VM . 2005 . “ On the delta-shock front problem ” . In Analytical Approaches to Multidimensional Balance Laws, Ch. 2 (Ed. O.S. Rozanova) , 45 – 88 . New York : Nova Science Publishers, Inc. .
   Google Scholar
• 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 .
   Web of Science ®Google Scholar
• J. Li and T. Zhang, On the initial-value problem for zero-pressure gas dynamics, hyperbolic problems: Theory, numerics, applications, Seventh International Conference in Zürich, February 1998, Birkhäuser Verlag, Basel, Boston, Berlin, 1999, pp. 629–640.
   Google Scholar
• Shelkovich , VM . 2008 . 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 ) : 73 – 146 . (English transl. in Russian Math. Surveys, 63:3 (2008), pp. 473–546)
   Google Scholar
• Shen , W and Zhang , T . 1999 . The Riemann problem for the transportation equations in gas dynamics . Mem. Amer. Math. Soc. , 137 ( 654 ) : 1 – 77 .
   Web of Science ®Google Scholar
• Yang , H . 1999 . Riemann problems for class of coupled hyperbolic systems of conservation laws . J. Diff. Eq. , 159 : 447 – 484 .
   Web of Science ®Google Scholar
• Panov , EYu and Shelkovich , VM . 2006 . δ′-Shock waves as a new type of solutions to systems of conservation laws . J. Diff. Eqns , 228 : 49 – 86 .
   Web of Science ®Google Scholar
• Shelkovich , VM . 2006 . The Riemann problem admitting δ-, δ′-shocks, and vacuum states (the vanishing viscosity approach) . J. Diff. Eqns , 231 : 459 – 500 .
   Web of Science ®Google Scholar
• Shelkovich , VM . 2006 . The Rankine–Hugoniot conditions and balance laws for δ-shocks . Fundament. Priklad. Mat. , 12 ( 6 ) : 213 – 229 .
   Google Scholar
• Shandarin , SF and Zeldovich , YaB . 1989 . The large-scale structure of the universe: Turbulence, intermittence, structures in self-gravitating medium . Rev. Mod. Phys. , 61 : 185 – 220 .
   Web of Science ®Google Scholar
• Zeldovich , YaB . 1970 . Gravitational instability: An approximate theory for large density perturbations . Astron. Astrophys. , 5 : 84 – 89 .
   Web of Science ®Google Scholar
• Kraiko , AN . 1979 . Discontinuity surfaces in medium without self-pressure . Prikl. Math. Mech. , 43 : 539 – 449 . (In Russian)
   Web of Science ®Google Scholar
• Kraiko , AN and Sulaimanova , SM . 1983 . Two-phase flows of a gas-particle mixture near impermeable surfaces with the formation of ‘sheets’ and ‘filaments’ . Priklad. Mat. Mekh. , 47 ( 4 ) : 619 – 630 . (In Russian)
   Google Scholar
• Kraiko , AN . 1982 . On two-phase flows model of gas and dispersed in it particles . Priklad. Mat. Mekh. , 46 ( 1 ) : 96 – 106 . (In Russian)
   Web of Science ®Google Scholar
• Kanwal , Ram P . 1998 . Generalized Functions: Theory and technique , Boston, Basel, Berlin : Birkhäuser .
   Google Scholar
• Gel'fand , IM and Shilov , GE . 1964 . “ Generalized Functions ” . In Properties and Operations , Vol. 1 , New York : Academic Press .
   Google Scholar