CONSTRUCTION OF THE 2D RIEMANN SOLUTIONS FOR A NONSTRICTLY HYPERBOLIC CONSERVATION LAW

Title & Authors
CONSTRUCTION OF THE 2D RIEMANN SOLUTIONS FOR A NONSTRICTLY HYPERBOLIC CONSERVATION LAW
Sun, Meina;

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.
Keywords
Riemann problem;generalized characteristic analysis;delta shock wave;vacuum state;
Language
English
Cited by
References
1.
S. Albeverio and V. M. Shelkovich, On the delta-shock front, in: Analytical Approaches to Multidimensional Balance Laws (Ed. O.S.Rozanova), pp.45-88, Nova Science Publishers, 2006.

2.
F. Bouchut, On zero pressure gas dynamics, in: Advances in kinetic theory and computing, 171-190, Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, 1994.

3.
A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl., vol. 20, Oxford University Press, Oxford, 2000.

4.
L. Guo, W. Sheng, and T. Zhang, The Two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal. 9 (2010), no. 2, 431-458.

5.
F. Huang and X. Yang, The two-dimensional Riemann problem for a class of systems of hyperbolic conservation law equations, Acta Math. Appl. Sinica 21 (1998), no. 2, 193-205.

6.
W. Hwang and W. B. Lindquist, The 2-dimensional Riemann problem for a 2 ${\times}$ 2 hyperbolic law, (I) Isotropic media, SIAM J. Math. Anal. 34 (2002), no. 2, 341-358

7.
W. Hwang and W. B. Lindquist, The 2-dimensional Riemann problem for a 2 ${\times}$ 2 hyperbolic law, (II) Anisotropic media, SIAM J. Math. Anal. 34 (2002), no. 2, 359-384.

8.
G. Lai, W. Sheng, and Y. Zheng, Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions, Discrete Contin. Dyn. Syst. 31 (2011), no. 2, 489-523.

9.
P. G. LeFloch, An existenceand uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations that change Type, IMA Vol. Math. Appl. 27 ed B. Keyfitz and M. Shearer, Berlin, Springer, 107-125, 1990.

10.
J. Li, T. Zhang, and S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 98, Longman Scientific and Technical, 1998.

11.
W. B. Lindquist, The scalar Riemann problem in two spatial dimensions: Piecewise smoothness of solutions and its breakdown, SIAM J. Math. Anal. 17 (1986), no. 5, 1178-1197.

12.
W. B. Lindquist, Construction of solutions for two-dimensional Riemann problems, Comput. Math. Appl. Part A 12 (1986), no. 4-5, 615-630.

13.
T. P. Liu and J. Smoller, On the vacuum state for isentropic gas dynamic equations, Adv. in Appl. Math. 1 (1980), no. 4, 345-359.

14.
M. Nedeljkov, Shadow waves: entropies and interactions for delta and singular shocks, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 487-537.

15.
M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008), no. 2, 1143-1157.

16.
V. M. Shelkovich, Singular solutions of ${\delta}$- and ${\delta}^1-shock$ wave type of systems of conservation laws, and transport and concentration processes, Uspekhi Mat. Nauk 63 (2008), no. 3(381), 73-146; translation in Russian Math. Surveys 63 (2008), no. 3, 473-546.

17.
C. Shen and M. Sun, 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 (2010), no. 12, 3024-3051.

18.
C. Shen and M. Sun, Interactions of delta shock waves for the transport equations with split delta functions, J. Math. Anal. Appl. 351 (2009), no 2, 747-755.

19.
C. Shen, M. Sun, and Z. Wang, Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws, Nonlinear Anal. 74 (2011), no. 14, 4754-4770.

20.
W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Amer. Math. Soc. 137 (1999), no. 654, viii+77 pp.

21.
W. Sun and W. Sheng, The non-selfsimilar Riemann problem for 2-D zero-pressure flow in gas dynamics, Chin. Ann. Math. Ser. B 28 (2007), no. 6, 701-708.

22.
D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws, (I): Four-J cases, J. Differential Equations 111 (1994), no. 2, 203-254.

23.
D. Tan and T. Zhang, Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws, (II): Initial data consists of some rarefaction, J. Differential Equations 111 (1994), no. 2, 255-283.

24.
D. Yoon and W. Hwang, Two-dimensional Riemann problem for Burgers equations, Bull. Korean Math. Soc. 45 (2008), no 1, 191-205.

25.
T. Zhang and Y. Zheng, Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal. 21 (1990), no. 3, 593-630.

26.
Y. Zheng, Systems of Conservation Laws, Birkhauser Verlag, 2001.