Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law

doi:10.1007/s42967-022-00245-7

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (3): 1180-1234.doi: 10.1007/s42967-022-00245-7

• ORIGINAL PAPER • Previous Articles Next Articles

Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law

Gui-Qin Qiu^1,2, Gao-Wei Cao^1,2, Xiao-Zhou Yang^1,2, Yuan-An Zhao³

1. University of Chinese Academy of Sciences, Beijing, 100049, China;
2. Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, 430071, Hubei, China;
3. Hubei Minzu University, Enshi, 445000, Hubei, China

Received:2022-02-18 Revised:2022-12-08 Online:2023-09-20 Published:2023-08-29
Contact: Xiao-Zhou Yang,E-mail:xzyang@wipm.ac.cn E-mail:guiqin_qiu@163.com;gwcao@wipm.ac.cn;xzyang@wipm.ac.cn;yuananzhao@hbmzu.edu.cn
Supported by:
In the year of 2022 that highly prestigious Prof. Tong Zhang is ninety years old, Xiao-zhou Yang and his co-authors submit this paper to celebrate Prof. Zhang’s 90th birthday and sincerely thank Prof. Zhang’s great guidance and help to Xiao-zhou Yang for over twenty years, Xiao-zhou Yang will also specially thank the long-term help from Prof. Yu-xi Zheng, Academican Ping Zhang, Prof. Wancheng Sheng, Prof. Jiequan Li, and Prof. Hanchun Yang, who are all very famous students of Prof. Zhang. Particularly, Xiao-zhou Yang will also thank Prof. Chi-Wang Shu for his benefit guidance. The research of Xiao-zhou Yang was supported in part by the NSFC (Grant No. 11471332). The research of Gao-wei Cao was supported in part by the NSFC (Grant No. 11701551).

Abstract

Abstract: For the two-dimensional (2D) scalar conservation law, when the initial data contain two different constant states and the initial discontinuous curve is a general curve, then complex structures of wave interactions will be generated. In this paper, by proposing and investigating the plus envelope, the minus envelope, and the mixed envelope of 2D non-selfsimilar rarefaction wave surfaces, we obtain and the prove the new structures and classifications of interactions between the 2D non-selfsimilar shock wave and the rarefaction wave. For the cases of the plus envelope and the minus envelope, we get and prove the necessary and sufficient criterion to judge these two envelopes and correspondingly get more general new structures of 2D solutions.

Key words: Riemann problem, Non-selfsimilar shock wave, Non-selfsimilar rarefaction wave, Envelope, Multi-dimensional conservation law

CLC Number:

35L65
35L67

Gui-Qin Qiu, Gao-Wei Cao, Xiao-Zhou Yang, Yuan-An Zhao. Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law[J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1180-1234.

TrendMD

References

[1] Ben-Artzi, M., Falcovitz, J., Li, J.Q.: Wave interactions and numerical approximation for two-dimensional scalar conservation laws. J. Comput. Fluid Dyn. 14(4), 401–418 (2006)
[2] Cao, G.W., Hu, K., Yang, X.Z.: Envelope and classification of global structures of solutions for a class of two-dimensional conservation laws. Acta Math. Appl. Sin. Engl. Ser. 32(3), 579–590 (2016)
[3] Cao, G.W., Xiang, W., Yang, X.Z.: Global structure of admissible solutions of multi-dimensional non-homogeneous scalar conservation law with Riemann-type data. J. Differential Equations 263(2), 1055–1078 (2017)
[4] Chen, G.Q., Li, D., Tan, D.: Structure of Riemann solutions for 2-dimensional scalar conservation laws. J. Differential Equations 127(1), 124–147 (1996)
[5] Chung, T., Hsiao, L.: The Riemann Problem and Interation of Waves in Gas Dynamics. Longman Scientific and Technical, London (1989)
[6] Conway, E., Smoller, J.: Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables. Comm. Pure Appl. Math. 19(1), 95–105 (1966)
[7] Guckenheimer, J.: Shocks and rarefcations in two space dimensions. Arch. Rational Mech. Anal. 59(3), 281–291 (1975)
[8] Kruzkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sb. 10(2), 217–243 (1970)
[9] Li, J.Q., Sheng, W.C., Zhang, T., Zheng, Y.X.: Two-dimensional Riemann problems: from scalar conservation laws to compressible Euler equations. Acta Math. Sci. 29(4), 777–802 (2009)
[10] Li, J.Q., Zhang, T., Yang, S.L.: The two-dimensional Riemann problems in gas dynamics. In: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 98. Longman Harlow, London (1998)
[11] Lindquist, W.B.: The scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown. SIAM J. Math. Anal. 17(5), 1178–1197 (1986)
[12] Lions, P.L., Perthame, P., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7(1), 169–191 (1994)
[13] Sheng, W.C.: Two-dimensional Riemann problem for scalar conservation laws. J. Differential Equations 183(1), 239–261 (2002)
[14] Sheng, W.C., Zhang, T.: A cartoon for the climbing ramp problem of a shock and von Neumann paradox. Arch. Ration. Mech. Anal. 184(2), 243–255 (2007)
[15] Volpert, A.I.: The space BV and quasilinear equations. Math. USSR Sb. 2(2), 225–267 (1967)
[16] Wagner, D.H.: The Riemann problem in two space dimensions for a single conservation law. SIAM J. Math. Anal. 14(3), 534–559 (1983)
[17] Yang, X.Z.: Multi-dimensional Riemann problem of scalar conservation law. Acta Math. Sci. 19(2), 190–200 (1999)
[18] Yang, X.Z., Wei, T.: New structures for non-selfsimilar solutions of multi-dimensional conservation laws. Acta Math. Sci. 29(5), 1182–1202 (2009)
[19] Yang, X.Z., Zhang, T.: Global smooth solution of multi-dimensional non-homogeneous conservation laws. Prog. Nat. Sci. 14(10), 855–862 (2004)
[20] Zhang, P., Zhang, T.: Generalized characteristic analysis and Guckenheimer structure. J. Differential Equations 152(2), 409–430 (1999)
[21] Zhang, T., Zheng, Y.X.: Two-dimensional Riemann problem for a single conservation law. Trans. Am. Math. Soc. 312(2), 589–619 (1989)
[22] Zheng, Y.X.: Systems of Conservation Laws: Two-Dimensional Riemann Problems. Birkhauser, Boston (2001)

Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 5

Recommended Articles

Metrics

Comments

[1]	Gui-Qiang G. Chen. Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1015-1052.
[2]	Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1097-1129.
[3]	Shiwei Li, Hanchun Yang. The Perturbed Riemann Problem for a Geometrical Optics System [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1148-1179.
[4]	R. Demattè, V. A. Titarev, G. I. Montecinos, E. F. Toro. ADER Methods for Hyperbolic Equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 369-402.
[5]	Steven Jöns, Claus, Dieter Munz. An Approximate Riemann Solver for Advection–Difusion Based on the Generalized Riemann Problem [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 515-539.