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Hyperbolic Conservation Laws and Related Analysis with Applications pp 113–142Cite as

Existence and Stability of Global Solutions of Shock Diffraction by Wedges for Potential Flow

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 49))

Abstract

We present our recent results on the mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the potential flow equation. The shock diffraction problem can be formulated as an initial-boundary value problem, which is invariant under self-similar scaling. Then, by employing its self-similar invariance, the problem is reduced to a boundary value problem for a first-order nonlinear system of partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It is further reformulated as a free boundary problem for a nonlinear degenerate elliptic system of first-order in a bounded domain with a boundary corner whose angle is bigger than π. A first global theory of existence and regularity has been established for this shock diffraction problem for the potential flow equation.

2010 Mathematics Subject Classification Primary: 35M10, 35M12, 35B65, 35L65, 35L70, 35J70, 76H05, 35L67, 35R35; Secondary: 35L15, 35L20, 35J67, 76N10, 76L05

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Acknowledgements

The research of Gui-Qiang G. Chen was supported in part by the National Science Foundation under Grant DMS-0807551, the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal Society–Wolfson Research Merit Award (UK). Wei Xiang was supported in part by the China Scholarship Council No. 2008631071 while visiting the University of Oxford and the Doctoral Program Foundation of the Ministry Education of China.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Gui-Qiang G. Chen & Wei Xiang

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Wei Xiang

Authors
  1. Gui-Qiang G. Chen
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  2. Wei Xiang
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Correspondence to Gui-Qiang G. Chen .

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Editors and Affiliations

  1. University of Oxford Mathematical Institute, Oxford, United Kingdom

    Gui-Qiang G. Chen

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Helge Holden

  3. University of Oslo Centre of Mathematics for Applications, Oslo, Norway

    Kenneth H. Karlsen

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Chen, GQ.G., Xiang, W. (2014). Existence and Stability of Global Solutions of Shock Diffraction by Wedges for Potential Flow. In: Chen, GQ., Holden, H., Karlsen, K. (eds) Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proceedings in Mathematics & Statistics, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39007-4_6

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