SOLUTIONS TO QUASILINEAR HYPERBOLIC CONSERVATION LAWS WITH INITIAL DISCONTINUITIES

Haiping NIU; Shu WANG

doi:10.1016/S0252-9602(17)30127-3

Acta mathematica scientia, Series B >

2018 , Vol. 38 >Issue 1: 203 - 219

DOI: https://doi.org/10.1016/S0252-9602(17)30127-3

Articles

SOLUTIONS TO QUASILINEAR HYPERBOLIC CONSERVATION LAWS WITH INITIAL DISCONTINUITIES

Haiping NIU ,
Shu WANG

Expand

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

Received date: 2016-12-07

Revised date: 2017-10-27

Online published: 2018-02-25

Supported by

The second author is supported by the National Natural Science Foundation of China (11371042, 1471028, 11601021) and the Beijing Natural Science Foundation (1142001).

Fold

Abstract

We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t < T^*.Then, when T^* ≤ t < T', we get a new shock wave between two rarefactions, and then, when t > T', another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t → ∞. The technique does not involve dimensional reduction or coordinate transformation.

Key words： singular structure; quasilinear hyperbolic equations; elementary wave; global solutions

Cite this article

Haiping NIU , Shu WANG . SOLUTIONS TO QUASILINEAR HYPERBOLIC CONSERVATION LAWS WITH INITIAL DISCONTINUITIES[J]. Acta mathematica scientia, Series B, 2018 , 38(1) : 203 -219 . DOI: 10.1016/S0252-9602(17)30127-3

References

[1] Bressan A. Hyperbolic Systems of Conservation Laws. London:Oxford University Press, 2000
[2] Chen G Q, Li D N, Tan D C. Structure of Riemann solution for 2-dimensional scalar conservation laws. J Differ Equ, 1996, 127(1):124-147
[3] Conway E, Smoller J A. Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables. Comm Pure Appl Math, 1966, 19:95-105
[4] Guckenheimer J. Shocks and rarefactions in two space Dimensions. Arch Rational Mech Anal, 1975, 59:281-291
[5] Hoff D. Locally Lipschitz of a single conservation law in several space viariable. J Differ Equ, 1981, 42(2):215-233
[6] Hoff D. The sharp from of Oleinic's entropy condition in several space variables. Trans Amer Math Soc, 1983, 276(2):707-714
[7] Kroner D, Rokyta M. Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J Numer Anal, 1994, 31(2):324-343
[8] Kruzkov S N. Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order. Soviet Math Dokl, 1969, 10:785-788
[9] Lax Peter D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. New York:Society for Industrial and Applied Mathematics, 1973
[10] Lindquist W B. The scalar Riemann problem in two spatial dimensions:piecewise somoothness of solutions and its breakdown. SIAM J Math Anal, 1986, 17:1178-1197
[11] Smoller J A. Shock Wave and Reaction-Diffusion Equation, 2nd Ed. New York:Springer-Verlag, 1999
[12] Szepessy A. Convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions. Math Comp, 1989, 188(53):527-545
[13] Vol'pert A I. The spaces BV and quasilinera equations. Math USSR-Sb, 1967, 2:225-267
[14] Wagner D. The Riemann problem in two space dimensions for a single conservation law. SIAM J Math Anal, 1982, 3:534-559
[15] Xu J C, Ying L A. Convergence of an explicit upwind finite element method to multi-dimensional conservation laws. J Comput Math, 2001, 19(1):87-100
[16] Yang X Z. Multi-dimensional Riemann problem of scalar conservation law. Acta Math Sci, 1999, 19(2):190-200
[17] Yang X Z. The singular structure of non-selfsimilar global solutions of n dimensional Burgers equation. Acta Math Appl Sin, Engl Ser, 2005, 21(3):505-518
[18] Yang X Z, Wei T. New structures for non-selfsimilar solutions of muti-dimensional conservation laws. Acta Math Sci, 2009, 29B(5):1182-1202
[19] Zhang P, Zhang T. Generalized characteristic analysis and Guckenheimer structure. J Differ Equ, 1999, 152(2):409-430
[20] Zhang T, Chen G Q. Some fundamental concepts for system of two special dimensional conservation laws. Acta Math Sci, 1986, 6:463-474
[21] Zhang T, Zheng Y X. Two-dimensional Riemann problem for a single conservation law. Trans Amer Math Soc, 1989, 312(2):589-619
[22] Zhang T, Zheng Y X. Conjecture on structure of solutions of Riemann problem for two-dimensional gas dynamics systems. SIAM J Math Anal, 1990, 21:593-630
[23] Zheng Y X. Two-Dimensional Riemann Problems. Boston:Birkhäuser, 86-107, 2001

Options

Outlines

模态框（Modal）标题

Abstract

Cite this article

References