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Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System

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Abstract

We are concerned with rigorous mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow, through the nonlinear wave system. This shock diffraction problem can be formulated as a boundary value problem for second-order nonlinear partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition. We establish a global theory of existence and optimal regularity for this shock diffraction problem. To achieve this, we develop several mathematical ideas and techniques, which are also useful for other related problems involving similar analytical difficulties.

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References

  1. Bae M., Chen G.-Q., Feldman M.: Regularity of solutions to regular shock reflection for potential flow. Invent. Math. 175, 505–543 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Bargman, V.: On nearly glancing reflection of shocks. Office Sci. Res. Develop. Rep. No. 5117 (1945)

  3. Canić S., Keyfitz B.L., Kim E.H.: A free boundary problem for a quasilnear degenerate elliptic equation: Transonic regular reflection of weak shocks. Comm. Pure Appl. Math. 55, 71–92 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Canić S., Keyfitz B.L., Kim E.H.: Free foundary problems for nonlinear wave systems: Mach stems for interacting shocks. SIAM J. Math. Anal. 37, 1947–1977 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen G.-Q., Feldman M.: Potential theory for shock reflection by large-angle wedges. Proce. Natl. Acade. Sci USA (PNAS) 102, 15368–15372 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Chen G.-Q., Feldman M.: Global solutions to shock reflection by large-angle wedges for potential flow. Ann. Math. 171, 1067–1182 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, G.-Q., Feldman, M.: Mathematics of Shock Reflection-Diffraction and von Neumann Conjectures, Monograph, Oxford, (2013)

  8. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Reprinting of the 1948 original. Applied Mathematical Sciences, Vol. 21. Springer, New York, 1976

  9. Fletcher C.H., Taub A.H., Bleakney W.: The Mach reflection of shock waves at nearly glancing incidence. Rev. Modern Phys. 23(3), 271–286 (1951)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Fletcher C.H., Weimer D.K., Bleakney W.: Pressure behind a shcok wave diffracted through a small angle. Phys. Rev. 78(5), 634–635 (1950)

    Article  ADS  Google Scholar 

  11. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd Edn. Springer, Berlin, 1983

  12. Kim E.H.: A global sub-sonic solution to an interacting transonic shock of the self-similar nonlinear wave equation. J. Differ. Equ. 248, 2906–2930 (2010)

    Article  ADS  MATH  Google Scholar 

  13. Lieberman G.: The Perron process applied to oblique derivative problems. Adv. Math. 55, 161–172 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lieberman G.: Regularized distance and its applications. Pac. J. Math. 117, 329–352 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lieberman G.: Mixed boundary value problems for elliptiv and parabolic differential equation of second order. J. Math. Anal. Appl. 113, 422–440 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lieberman G.: Oblique derivative problems in Lipschitz domains, II. Discontinuous boundary data. J. Reine Angew. Math. 389, 1–21 (1988)

    MATH  MathSciNet  Google Scholar 

  17. Lieberman G.: Optimal Hölder regularity for mixed boundary value problems. J. Math. Anal. Appl. 143, 572–586 (1989)

    Article  MathSciNet  Google Scholar 

  18. Lighthill M.J. (1949) The diffraction of blast I. Proc. Roy. Soc. 198A, 454–470

    Google Scholar 

  19. Lighthill, M.J.: The diffraction of blast II. Proc. Roy. Soc. 200A, 554–565 (1950)

    Google Scholar 

  20. Serre D.: Multidimensional shock interaction for a Chaplygin gas. Arch. Ration. Mech. Anal. 191, 539–577 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Whitham, G.B.: Linear and Nonlinear Waves. Reprint of the 1974 original. A Wiley-Interscience Publication. Wiley, New York, 1999

  22. Zheng, Y.: Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sinica. (English Ser) 22, 177–210 (2006)

    Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Gui-Qiang Chen, Xuemei Deng & Wei Xiang

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

    Gui-Qiang Chen & Wei Xiang

  3. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Gui-Qiang Chen

  4. College of Science, China Three Gorges University, Yichang, Hubei, 443002, China

    Xuemei Deng

  5. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China

    Xuemei Deng

Authors
  1. Gui-Qiang Chen
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  2. Xuemei Deng
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  3. Wei Xiang
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Correspondence to Gui-Qiang Chen.

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Communicated by C. Dafermos

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Chen, GQ., Deng, X. & Xiang, W. Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System. Arch Rational Mech Anal 211, 61–112 (2014). https://doi.org/10.1007/s00205-013-0681-1

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  • DOI: https://doi.org/10.1007/s00205-013-0681-1

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