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High Speed Flight and Partial Differential Equations

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Abstract

Aircraft comes out at the beginning of the last century. Accompanied by the progress of high speed flight the theory of partial differential equations has been greatly developed. This paper gives a brief review on the history of applications of partial differential equations to the study of supersonic flows arising in high speed flight.

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References

  1. Bers, L., Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley and Sons, Inc., New York, 1958.

    MATH  Google Scholar 

  2. Chen, G. Q. and Fang, B. X., Stability of transonic shocks in steady supersonic flow past multidimensional wedges, Adv. Math., 314, 2017, 493–539.

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, G. Q. and Feldman, M., Global solutions of shock reflection by large-angle wedges for potential flow, Ann. Math., 172, 2010, 1067–1182.

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, S. X., Existence of local solution to supersonic flow around a three dimensional wing, Adv. Appl. Math., 13, 1992, 273–304.

    Article  MATH  Google Scholar 

  5. Chen, S. X., Existence of stationary supersonic flow past a pointed body, Archive Rat. Mech. Anal., 156, 2001, 141–181.

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, S. X., Study of multidimensional systems of conservation laws: Problems, Difficulties and Progress, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, 1884–1900.

  7. Chen, S. X., Mathematical Analysis of Shock Wave Reflection, Springer Press and Shanghai Sci. Tech. Pub., Shanghai, 2020.

    Book  MATH  Google Scholar 

  8. Chen, S. X. and Fang, B. X., Stability of transonic shocks in supersonic flow past a wedge, Jour. Diff. Eqs., 233, 2007, 105–135.

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, S. X. and Li, D. N., Supersonic flow past a symmetric curved cone, Indiana Univ. Math. Jour., 49, 2000, 1411–1435.

    Article  MATH  Google Scholar 

  10. Chen, S. X. and Li, D. N., The existence and stability of conic shock waves, Jour. Math. Anal. Appl., 277, 2003, 512–532.

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, S. X. and Li, D. N., Conical shock waves in supersonic flow, Journal of Differential Equations, 269, 2020, 595–611.

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, S. X., Xin, Z. P. and Yin, H. C., Global shock waves for the supersonic flow past a perturbed cone, Comm. Math. Phys., 228, 2002, 47–84.

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, S. X. and Yi, C., Global solutions for supersonic flow past a delta wing, SIAM Math. Anal., 47, 2015, 80–126.

    Article  MATH  MathSciNet  Google Scholar 

  14. Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948.

    MATH  Google Scholar 

  15. Gu, C. H., A method for solving the supersonic flow past a curved wedge, Fudan Journal (Natur. Sci.), 7, 1962, 11–14.

    Google Scholar 

  16. Gu, C. H., Li, T. T. and Hou, Z. Y., The Cauchy problem of hyperbolic systems with discontinuous initial values I,II,III, Acta Math. Sinica, 4, 1961, 314–323, 324–327, 5, 1962, 132–143.

    Google Scholar 

  17. Fang, B. X., Stability of transonic shocks for the full Euler system in supersonic flow past a wedge, Math. Mach. Appl. Sci., 29, 2006, 1–26.

    MATH  MathSciNet  Google Scholar 

  18. Lax, P. D., Hyperbolic system of conservation laws, Comm. Pure Appl. Math., 10, 1957, 537–566.

    Article  MATH  MathSciNet  Google Scholar 

  19. Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conf. Board. Math. Sci., 11 SIAM, 1973.

  20. Li, J., Witt, I. and Yin, H. C., On the global existemce and stability of a multidimensional supersonic conic shock wave, Comm. Math. Phys., 329, 2014, 609–640.

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, T.-T. and Yu, W.-C., Some existence theorems for quasilinear hyperbolic systems of partial differential equations in two independent variables, I, II, Scienia Sinica, 4, 1964, 529–550, 551–562.

    Google Scholar 

  22. Li, T.-T. and Yu, W.-C., Boundry Value Problems for Quasi-linear Hyperbolic Systems, Duke Univ. Math. Ser. V., Duke Unir. Press, Durhan, 1985.

    Google Scholar 

  23. Lien, W. C. and Liu, T. P., Nonlinear stability of a self-similar 3-D gas flow, Comm. Math. Phys., 304, 1999, 524–549.

    MathSciNet  Google Scholar 

  24. Majdam, A. J., The stability of multidimensional shock front, The existence of multidimensional shock front, Memoirs Amer. Math. Soc., 275, 281, 1983.

  25. Schaeffer, D. G., Supersonic flow past a nearly straight wedge, Duke Math. J., 43, 1976, 637–670.

    Article  MATH  MathSciNet  Google Scholar 

  26. Xin, Z. P. and Yin, H. C., Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. (Singap.), 4, 2006, 101–132.

    Article  MATH  MathSciNet  Google Scholar 

  27. Xu, G. and Yin, H. C., Global multidimensional transonic conic shock wave for the perturbed supersonic flow past a cone, SIAM J. Math. Anal., 41, 2009, 178–218.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Shuxing Chen.

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Chen, S. High Speed Flight and Partial Differential Equations. Chin. Ann. Math. Ser. B 43, 855–868 (2022). https://doi.org/10.1007/s11401-022-0363-0

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  • DOI: https://doi.org/10.1007/s11401-022-0363-0

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