NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

doi:10.1016/S0252-9602(09)60096-5

数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (5): 1182-1202.doi: 10.1016/S0252-9602(09)60096-5

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

杨小舟, 魏涛

1.Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, West No.30, Xiao Hong Shan, Wuhan 430071, China；2.Department of Mathematics, Shantou University, Shantou 515063, China

收稿日期:2007-08-19 出版日期:2009-09-20 发布日期:2009-09-20
基金资助:
Sponsored by the National Natural Science Foundation of China (10671116, 10871199, and 10001023), Hou Yingdong Fellowship (81004), The China Scholarship Council, Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, Natural Science Foundation of Guangdong (06027210 and 000804) and Natural Science Foundation of Guangdong Education Bureau (200030)

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

YANG Xiao-Zhou, WEI Tao

1.Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, West No.30, Xiao Hong Shan, Wuhan 430071, China；2.Department of Mathematics, Shantou University, Shantou 515063, China

Received:2007-08-19 Online:2009-09-20 Published:2009-09-20
Supported by:
Sponsored by the National Natural Science Foundation of China (10671116, 10871199, and 10001023), Hou Yingdong Fellowship (81004), The China Scholarship Council, Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, Natural Science Foundation of Guangdong (06027210 and 000804) and Natural Science Foundation of Guangdong Education Bureau (200030)

摘要/Abstract

摘要：

In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure.

This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of two-dimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u₊>0, while explicit non-selfsimilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u₊<0.

The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension.

关键词: non selfsimilar, shock wave, rarefaction wave, envelope, multi dimensional conservation laws

Abstract:

Key words: non selfsimilar, shock wave, rarefaction wave, envelope, multi dimensional conservation laws

中图分类号:

35L65

杨小舟, 魏涛. NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS[J]. 数学物理学报(英文版), 2009, 29(5): 1182-1202.

YANG Xiao-Zhou, WEI Tao. NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS[J]. Acta mathematica scientia,Series B, 2009, 29(5): 1182-1202.

参考文献

[1] Chang T, Hsiao L. Riemann Problem and Interaction of Waves in Gas Dynamics. Pitman Monogr, Jurveys Pure Appl Math, Vol 41. Essex: Longman, 1989

[2] Chen Guiqiang, Li Dening, Tan Dechun. Structure of Riemann solution for 2-dimensional scalar conservation laws. J Differential Equations, 1996, 127(1): 124--147

[3] Conway E, Smoler J. Global solution of the Cauchy problem for quasilinear first-order equations in several 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] Hsiao L, Kilingenberg C. The structure of the solution for the two-dimensional Riemann problem. preprint, Heidelberg Univ, 1986

[6] Kruzkov S N. Order quasilinear equations with several independent variables. Math USSR Sb, 1970, 10: 271--243

[7] Lax Peter D. Hyperbolic systems of conservation laws and the mathematical theory of dhock waves. Society for Industrial and Applied Mathematics, 1973

[8] Lindquist W B. The sc'dar Riemann problem in two spatial dimensions: Piecewise smoothness of solutions. SIAM J Math Anal, 1986, 17: 1178--1197. MR0853523 (87j:35064)

[9] Sheng Wancheng. Two-dimensional Riemann problem for scalar conservation laws. J Differential Equations, 2002, 183(1): 239--261

[10] Wagner D. The Riemann problem in two space dimensions for a single conservation law. SIAM J Math Anal, 1983, 38: 534--559. MR0697528 (84f:35002)

[11] Yang Xiaozhou. Multi-Dimensional Riemann problem of scalar conservation law. Acta Math Sci, 1990, 10(2): 190--200

[12] Yang Xiaozhou, Huang Feimin. Two dimensional Riemann problem of simplified Euler equation. Chinese Science Bulletin, 1998, 43(6): 441--444

[13] Zhang T, Zheng Y X. Two-dimensional Riemann problem for a single conservation law. Trans Amer Math Soc, 1989, 312(2): 589--619

[14] Zhang Peng, Zhang Tong. Generalized characteristic analysis and Guckenheimer structure. J Differential Equations, 1999, 152(2): 409--430

[15] Zheng Yuxi. Two-dimensional Riemann Problems. Boston: Birkhäuser, 2001. 86--107

[16] Bressan Alberto. Hyperbolic Systems of Conservation Laws. Oxford University Press, 2000

[17] Smoller J. Shock Wave and Reaction Diffusion Equation. 2nd ed. New York: Springer-Verlag, 1999

[18] Yang Xiaozhou. The singular structure of non-selfsimilar global solutions of n dimensional Burgers equation. Acta Math Appl Sin, Engl Ser, 2005, 21(3): 505--518

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

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