Shuxing Chen - Mathematical Analysis of Shock Wave Reflection

Series in Contemporary Mathematics (SCM), featuring high-quality mathematical monographs, is to presents original and systematic findings from the fields of pure mathematics, applied mathematics and math-related interdisciplinary subjects. It has a history of over fifty years since the first title was published by Shanghai Scientific & Technical Publishers in 1963. Professor HUA Luogeng (Lo-Keng Hua) served as Editor-in-Chief of the first editorial board, while Professor SU Buqing acted as Honorary Editor-in-Chief and Professor GU Chaohao as Editor-in-Chief of the second editorial board since 1992. Now the third editorial board is established and Professor LI Tatsien assumes the position of Editor-in-Chief. The series has already published twenty-six monographs in Chinese, and among the authors are many distinguished Chinese mathematicians, including the following members of the Chinese Academy of Sciences: SU Buqing, GU Chaohao, LU Qikeng, ZHANG Gongqing, CHEN Hanfu, CHEN Xiru, YUAN Yaxiang, CHEN Shuxing etc. The monographs have systematically introduced a number of important research findings which not only play a vital role in China, but also exert huge influence all over the world. Eight of them have been translated into English and published abroad. The new editorial board will inherit and carry forward the former traditions and strengths of the series, and plan to further reform and innovation in terms of internalization so as to improve and ensure the quality of the series, extend its global influence, and strive to forge it into an internationally significant series of mathematical monographs.

More information about this series at http://www.springer.com/series/13634 Editor-in-Chief

Tatsien Li

Series Editors

Philippe G. Ciarlet, Jean-Michel Coron, Weinan E,

Jianshu Li, Jun Li, Tatsien Li, Fanghua Lin,

Zhi-ming Ma, Andrew J. Majda, Cdric Villani,

Ya-xiang Yuan, Weiping Zhang

Jointly published with Shanghai Scientific and Technical Publishers

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In the motion of continuous media, such as compressible fluid, the occurrence and propagation of shock waves are common physical phenomena. For instance, the detonation of explosives in a continuous medium will cause an shock wave propagating starting from the source of the explosion; a fast flying projectile with supersonic speed always produces a shock wave ahead of the projectile, moving with it together. Physically, the shock wave is a very thin layer in the medium, and its characteristic feature is that the state of the medium in this thin layer changes rapidly. Then the parameters describing the medium, such as velocity, density, pressure, and temperature, etc., generally may have significant change from the one side of the layer to the other side of it. Mathematically, the shock wave is often described by a surface with zero width, and the parameters of the fluid are discontinuous on this surface. The occurrence of shock waves brings great influence to the physical state of the medium around it. Particularly, in the case when a shock hits an obstacle and then is reflected, the reflection is often powerful and produces severe damage. Therefore, it is crucial to deeply understand and give great concern on the occurrence, propagation, and reflection of shock waves. Obviously, since the obstacles could be in various way, the structure of shock waves and the flow field caused by the reflection of shocks would be quite complicated. Consequently, precisely understanding the process of shock reflection and the resulting effect is very important and rather difficult.

Generally, their are three ways to study various problems in fluid dynamics: experiment investigation, numerical computation, and theoretical analysis. The theoretical analysis, especially the mathematical analysis, often predicts physical phenomena or offers qualitative characters to observed phenomena, the numerical computation offers required quantitative results in engineering technology, and the experiment investigation gives verification of obtained results or established conclusions, and occasionally finds new phenomena to raise new research topics. In any case, the theoretical analysis are indispensable for either numerical computation or experiment investigation. For instance, rigorous theoretical analysis points out that the flow parameters on both sides of any shock wave should satisfy Rankine-Hugoniot conditions and entropy condition, then these conditions have become basic rule for numerical computation in compressible flow involving shock waves. Since the recent development of engineering technology requires more precise and accurate numerical results, then more efficient mathematical tools, particularly the theory of partial differential equations, are expected to play their role. However, we should say that though the theory of partial differential equations developed rapidly in recent decades, the application of the theory to the problems involving shock waves is far from enough and anticipated. The situation reminds us the words written by R. Courant and K. O. Friedrichs in their book Supersonic flow and shock waves [1]: