Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Two-dimensional Riemann problem for pressureless gas dynamics equations with functional solutions

Author: Jiaxin Hu
Journal: Quart. Appl. Math. 58 (2000), 251-264
MSC: Primary 35L40; Secondary 35Q35, 76N15
DOI: https://doi.org/10.1090/qam/1753398
MathSciNet review: MR1753398
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Abstract: The Riemann problem for a two-dimensional hyperbolic system of pressureless gas dynamics equations is considered here. Riemann solutions are constructed for any piecewise-constant initial data having two discontinuity rays with the origin as vertex. Non-classical waves (labelled as Dirac-shock waves) appear in some solutions. The entropy conditions for the Dirac-shock waves are given. Uniqueness of Riemann solutions containing Dirac-shock waves fails.

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  • [1] R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Applied Mathematical Sciences, Vol. 21, Springer-Verlag, New York, 1976 MR 0421279
  • [2] X. Q. Ding, On a non-strictly hyperbolic system, preprint, Dept. of Math., University of Jyvaskyla, Finland, No. 167, 1993
  • [3] X. Q. Ding and Z. Wang, Existence and uniqueness of discontinuous solutions defined by Lebesgue-Stieltjes integral, Sci. China Ser. A 39, 807-819 (1996) MR 1417888
  • [4] J. X. Hu, One-dimensional Riemann problem for the equations of constant pressure fluid dynamics with functional solutions by the viscosity method, Acta Applicandae Mathematicae 55, 209-229 (1999) MR 1681816
  • [5] J. X. Hu, A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws, Quart. Appl. Math. LV, 361-373 (1997) MR 1447583
  • [6] D. J. Korchinski, Solution of a Riemann problem for a $ 2 \times 2$ system of conservation laws possessing no classical weak solution, Ph.D. thesis, Adelphi University, 1977 MR 2626928
  • [7] D. C. Tan, T. Zhang, and Y. X. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112, 1-32 (1994) MR 1287550
  • [8] T. Zhang and G. Q. Chen, Some fundamental concepts about system of two spatial dimensional conservation laws, Acta Math. Sci. 6, 463-474 (1986) MR 924036
  • [9] J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33, 1-16 (1996) MR 1377240
  • [10] E. Weinan, Yu. G. Rykov, and Ya. G. Sinai, 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 (1996) MR 1384139

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DOI: https://doi.org/10.1090/qam/1753398
Article copyright: © Copyright 2000 American Mathematical Society

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