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On Two-Dimensional Sonic-Subsonic Flow

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Abstract

A compensated compactness framework is established for sonic-subsonic approximate solutions to the two-dimensional Euler equations for steady irrotational flows that may contain stagnation points. Only crude estimates are required for establishing compactness. It follows that the set of subsonic irrotational solutions to the Euler equations is compact; thus flows with sonic points over an obstacle, such as an airfoil, may be realized as limits of sequences of strictly subsonic flows. Furthermore, sonic-subsonic flows may be constructed from approximate solutions. The compactness framework is then extended to self-similar solutions of the Euler equations for unsteady irrotational flows.

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References

  1. Ball, J.M. A version of the fundamental theorem for Young measures. Lecture Notes in Phys. 344, Berlin: Springer, 1989, pp. 207–215

  2. Bers L. (1954). Results and conjectures in the mathematical theory of subsonic and transonic gas flows. Comm. Pure Appl. Math. 7: 79–104

    MATH  MathSciNet  Google Scholar 

  3. Bers L. (1954). Existence and uniqueness of a subsonic flow past a given profile. Comm. Pure Appl. Math. 7: 441–504

    MATH  MathSciNet  Google Scholar 

  4. Bers L. (1958). Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Chapman & Hall, Ltd., New York, John Wiley & Sons, Inc., London

    MATH  Google Scholar 

  5. Chen, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 6, 75–120 (1986) (in English); 8, 243–276 (1988) (in Chinese)

  6. Chen, G.-Q.: Compactness Methods and Nonlinear Hyperbolic Conservation Laws. AMS/IP Stud. Adv. Math. 15, Providence, RI: AMS, 2000, pp. 33–75

  7. Chen, G.-Q.: Euler Equations and Related Hyperbolic Conservation Laws. In: Handbook of Differential Equations, Vol. 2, edited by C. M. Dafermos, E. Feireisl, Amsterdam: Elsevier Science B.V, pp. 1–104

  8. Chen G.-Q. and Frid H. (1999). Divergence-measure fields and hyperbolic conservation laws. Arch. Rational Mech. Anal. 147: 89–118

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Chen, G.-Q., LeFloch, Ph.: Compressible Euler equations with general pressure law. Arch. Rational Mech. Anal. 153, 221–259 (2000); Existence theory for the isentropic Euler equations. Arch. Rational Mech. Anal. 166, 81–98 (2003)

  10. Courant R. and Friedrichs K.O. (1962). Supersonic Flow and Shock Waves. Springer-Verlag, New York

    Google Scholar 

  11. Dafermos C.M. (2005). Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, Berlin

    MATH  Google Scholar 

  12. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I)–(II). Acta Math. Sci. 5, 483–500, 501–540 (1985) (in English); 7, 467–480 (1987), 8, 61–94 (1988) (in Chinese)

  13. DiPerna R.J. (1983). Convergence of viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91: 1–30

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. DiPerna R.J. (1983). Convergence of approximate solutions to conservation laws. Arch. Rational Mech. Anal. 82: 27–70

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. DiPerna R.J. (1985). Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc. 292: 383–420

    Article  MathSciNet  Google Scholar 

  16. Dong, G.-C.: Nonlinear Partial Differential Equations of Second Order. Translations of Mathematical Monographs, 95, Providence, RI: American Mathematical Society, 1991

  17. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS-RCSM, 74, Providence, RI: AMS, 1990

  18. Finn, R.: On the flow of a perfect fluid through a polygonal nozzle, I, II. Proc. Nat. Acad. Sci. USA. 40, 983–985, 985–987 (1954)

    Google Scholar 

  19. Finn R. (1954). On a problem of type, with application to elliptic partial differential equations. J. Rational Mech. Anal. 3: 789–799

    MathSciNet  Google Scholar 

  20. Finn R. and Gilbarg D. (1957). Asymptotic behavior and uniquenes of plane subsonic flows. Comm. Pure Appl. Math. 10: 23–63

    MATH  MathSciNet  Google Scholar 

  21. Finn R. and Gilbarg D. (1957). Three-dimensional subsonic flows and asymptotic estimates for elliptic partial differential equations. Acta Math. 98: 265–296

    Article  MATH  MathSciNet  Google Scholar 

  22. Finn R. and Gilbarg D. (1958). Uniqueness and the force formulas for plane subsonic flows. Trans. Amer. Math. Soc. 88: 375–379

    Article  MATH  MathSciNet  Google Scholar 

  23. Gilbarg D. (1953). Comparison methods in the theory of subsonic flows. J. Rational Mech. Anal. 2: 233–251

    MathSciNet  Google Scholar 

  24. Gilbarg D. and Serrin J. (1955). Uniqueness of axially symmetric subsonic flow past a finite body. J. Rational Mech. Anal. 4: 169–175

    MathSciNet  Google Scholar 

  25. Gilbarg D. and Shiffman M. (1954). On bodies achieving extreme values of the critical Mach number. I. J. Rational Mech Anal. 3: 209–230

    MathSciNet  Google Scholar 

  26. Lax P.D. (1973). Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia

    MATH  Google Scholar 

  27. Lions P.-L., Perthame B. and Souganidis P. (1996). Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49: 599–638

    Article  MATH  MathSciNet  Google Scholar 

  28. Lions P.-L., Perthame B. and Tadmor E. (1994). Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163: 169–172

    Article  MathSciNet  Google Scholar 

  29. Morawetz C.S. (1982). The mathematical approach to the sonic barrier. Bull Amer. Math. Soc. (N.S.) 6: 127–145

    Article  MATH  MathSciNet  Google Scholar 

  30. Morawetz C.S. (1985). On a weak solution for a transonic flow problem. Comm. Pure Appl. Math. 38: 797–818

    MATH  MathSciNet  Google Scholar 

  31. Morawetz C.S. (1995). On steady transonic flow by compensated compactness. Methods Appl. Anal. 2: 257–268

    MATH  MathSciNet  Google Scholar 

  32. Morawetz C.S. (2004). Mixed equations and transonic flow. J. Hyper. Diff. Eqs. 1: 1–26

    Article  MATH  MathSciNet  Google Scholar 

  33. Murat F. (1978). Compacite par compensation. Ann. Suola Norm. Pisa (4) 5: 489–507

    MATH  MathSciNet  Google Scholar 

  34. Serre, D.: Systems of Conservation Laws, Vols. 1–2, Cambridge: Cambridge University Press, 1999, 2000

  35. Shiffman M. (1952). On the existence of subsonic flows of a compressible fluid. J. Rational Mech. Anal. 1: 605–652

    MathSciNet  Google Scholar 

  36. Tartar L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV, Res. Notes in Math. 39, Boston-London: Pitman, 1979, pp. 136–212

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Authors and Affiliations

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

    Gui-Qiang Chen

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Constantine M. Dafermos

  3. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Marshall Slemrod

  4. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Dehua Wang

Authors
  1. Gui-Qiang Chen
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  2. Constantine M. Dafermos
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  3. Marshall Slemrod
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  4. Dehua Wang
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Correspondence to Gui-Qiang Chen.

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Communicated by P. Constantin.

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Chen, GQ., Dafermos, C.M., Slemrod, M. et al. On Two-Dimensional Sonic-Subsonic Flow. Commun. Math. Phys. 271, 635–647 (2007). https://doi.org/10.1007/s00220-007-0211-9

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