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Construction of the 2D Riemann solutions for a nonstrictly hyperbolic conservation law

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In this note, we consider the Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws. Without the restriction that each jump of the initial data projects one planar elementary wave, six topologically distinct solutions are constructed by applying the generalized characteristic analysis method, in which the delta shock waves and the vacuum states appear. Moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct global solutions.

Abstract

In this note, we consider the Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws. Without the restriction that each jump of the initial data projects one planar elementary wave, six topologically distinct solutions are constructed by applying the generalized characteristic analysis method, in which the delta shock waves and the vacuum states appear. Moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct global solutions.

참고문헌 (24)

  1. S. Albeverio, "On the delta-shock front, in: Analytical Approaches to Multidimensional Balance Laws", pp.45~88 (2006), Nova Science Publishers
  2. F. Bouchut, "On zero pressure gas dynamics, in: Advances in kinetic theory and computing", pp.171~190 (1994), World Sci. Publ.
  3. A. Bressan, "Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem" (2000), Oxford University Press
  4. L. Guo, "Two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system", Commun. Pure Appl. Anal, 9(2), pp.431~458 (2010)
  5. F. Huang, "The two-dimensional Riemann problem for a class of systems of hyperbolic conservation law equations", Acta Math. Appl. Sinica, 21(2), pp.193~205 (1998)
  6. W. Hwang, "The 2-dimensional Riemann problem for a 2 × 2 hyperbolic law, (I) Isotropic media, (II) Anisotropic media", SIAM J. Math. Anal, 34(2), pp.341~358 (2002)
  7. G. Lai, "Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions", Discrete Contin. Dyn. Syst, 31(2), pp.489~523 (2011)
  8. P. G. LeFloch, "An existenceand uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations that change Type", pp.107~125 (1990), Springer
  9. J. Li, "The Two-Dimensional Riemann Problem in Gas Dynam- ics, Pitman Monographs and Surveys in Pure and Applied Mathematics" (1998), Longman Scientific and Technical
  10. W. B. Lindquist, "The scalar Riemann problem in two spatial dimensions: Piecewise smoothness of solutions and its breakdown", SIAM J. Math. Anal, 17(5), pp.1178~1197 (1986)
  11. W. B. Lindquist, "Construction of solutions for two-dimensional Riemann problems", Comput. Math. Appl. Part A, 12(4), pp.615~630 (1986)
  12. T. P. Liu, "On the vacuum state for isentropic gas dynamic equations", Adv. in Appl. Math, 1(4), pp.345~359 (1980)
  13. M. Nedeljkov, "Shadow waves: entropies and interactions for delta and singular shocks", Arch. Ration. Mech. Anal, 197(2), pp.487~537 (2010)
  14. M. Nedeljkov, "Interactions of delta shock waves in a strictly hyperbolic system of conservation laws", J. Math. Anal. Appl, 344(2), pp.1143~1157 (2008)
  15. V. M. Shelkovich, "Singular solutions of δ- and δ0-shock wave type of systems of conser- vation laws, and transport and concentration processes", Uspekhi Mat. Nauk, 63(3), pp.73~146 (2008)
  16. C. Shen, "Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model", J. Differential Equations, 249(12), pp.3024~3051 (2010)
  17. C. Shen, "Interactions of delta shock waves for the transport equations with split delta functions", J. Math. Anal. Appl, 351(2), pp.747~755 (2009)
  18. C. Shen, "Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws", Nonlinear Anal, 74(14), pp.4754~4770 (2011)
  19. W. Sheng, "The Riemann problem for the transportation equations in gas dynamics", Mem. Amer. Math. Soc, 137(654) (1999)
  20. W. Sun, "The non-selfsimilar Riemann problem for 2-D zero-pressure flow in gas dynamics", Chin. Ann. Math. Ser. B, 28(6), pp.701~708 (2007)
  21. D. Tan, "-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws, (I): Four-J cases; (II): Initial data consists of some rar- efaction", J. Differential Equations, 111(2), pp.203~254 (1994)
  22. 윤대기, "Two-Dimensional Riemann Problem for Burgers' Equation", 대한수학회보, 45(1), pp.191~205 (2008), 대한수학회
  23. T. Zhang, "Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems", SIAM J. Math. Anal, 21(3), pp.593~630 (1990)
  24. Y. Zheng, "Systems of Conservation Laws" (2001), Birkhauser Verlag

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