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.
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Acknowledgements
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).
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This paper is dedicated to the 90th Birthday of Professor Tong Zhang.
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Qiu, GQ., Cao, GW., Yang, XZ. et al. Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law. Commun. Appl. Math. Comput. 5, 1180–1234 (2023). https://doi.org/10.1007/s42967-022-00245-7
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DOI: https://doi.org/10.1007/s42967-022-00245-7
Keywords
- Riemann problem
- Non-selfsimilar shock wave
- Non-selfsimilar rarefaction wave
- Envelope
- Multi-dimensional conservation law