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

Hu, Jiaxin

doi:10.1090/qam/1753398

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.

R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York-Heidelberg, 1976. Reprinting of the 1948 original; Applied Mathematical Sciences, Vol. 21. MR 0421279

X. Q. Ding, On a non-strictly hyperbolic system, preprint, Dept. of Math., University of Jyvaskyla, Finland, No. 167, 1993

Xiaxi Ding and Zhen Wang, Existence and uniqueness of discontinuous solutions defined by Lebesgue-Stieltjes integral, Sci. China Ser. A 39 (1996), no. 8, 807–819. MR 1417888
Jiaxin Hu, One-dimensional Riemann problem for equations of constant pressure fluid dynamics with measure solutions by the viscosity method, Acta Appl. Math. 55 (1999), no. 2, 209–229. MR 1681816, DOI https://doi.org/10.1023/A%3A1006101529302
Jiaxin Hu, A limiting viscosity approach to Riemann solutions containing delta-shock waves for nonstrictly hyperbolic conservation laws, Quart. Appl. Math. 55 (1997), no. 2, 361–373. MR 1447583, DOI https://doi.org/10.1090/qam/1447583
Dennis James Korchinski, SOLUTION OF A RIEMANN PROBLEM FOR A 2 X 2 SYSTEM OF CONSERVATION LAWS POSSESSING NO CLASSICAL WEAK SOLUTION, ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)–Adelphi University. MR 2626928
De Chun Tan, Tong Zhang, and Yu Xi Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), no. 1, 1–32. MR 1287550, DOI https://doi.org/10.1006/jdeq.1994.1093
Tong Chang and Gui Qiang Chen, Some fundamental concepts about system of two spatial dimensional conservation laws, Acta Math. Sci. (English Ed.) 6 (1986), no. 4, 463–474. MR 924036, DOI https://doi.org/10.1016/S0252-9602%2818%2930506-X
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 (1996), no. 1, 1–16. MR 1377240, DOI https://doi.org/10.1137/0733001
Weinan E, 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 (1996), no. 2, 349–380. MR 1384139