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The asymptotic solution of singularly perturbed Dirichlet problems with applications to the study of incompressible flows at high Reynolds number

Part II: Applications

Part of the Lecture Notes in Mathematics book series (LNM,volume 942)

Keywords

  • Boundary Layer
  • High Reynolds Number
  • Incompressible Flow
  • Singularly Perturb
  • Singular Perturbation Problem

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References

  1. H. Amann, Existence and Multiplicity Theorems for Semilinear Elliptic Boundary Value Problems, Math. Z. 150(1976), 281–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ. Press, 1970.

    Google Scholar 

  3. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. II, Interscience, New York, 1962.

    MATH  Google Scholar 

  4. W. Eckhaus, Boundary Layers in Linear Elliptic Singular Perturbation Problems, SIAM Rev. 14(1972), 225–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. W. Eckhaus and E. M. de Jager, Asymptotic Solutions of Singular Perturbation Problems for Linear Differential Equations of Elliptic Type, Arch. Rational Mech. Anal. 23(1966), 26–86.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. J. Grasman, On the Birth of Boundary Layers, Math. Centre Tract no. 36, Math. Centrum, Amsterdam, 1971.

    MATH  Google Scholar 

  7. F. A. Howes, Singularly Perturbed Semilinear Elliptic Boundary Value Problems, Comm. in Partial Differential Equations 4(1979), 1–39.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. F. A. Howes, Some Singularly Perturbed Nonlinear Boundary Value Problems of Elliptic Type, Proc. Conf. Nonlinear P.D.E.'s in Engrg. and Applied Sci., ed. by R. L. Sternberg, Marcel Dekker, New York, 1980, pp. 151–166.

    Google Scholar 

  9. F. A. Howes, Perturbed Boundary Value Problems Whose Reduced Solutions are Non-smooth, Indiana U. Math. J. 30(1981), 267–280.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. S. Kamin, On Equations of Elliptic and Parabolic Type with a Small Parameter Multiplying the Highest Derivatives (in Russian), Mat. Sbornik 31(1952), 703–708.

    MathSciNet  Google Scholar 

  11. M. Krzyzanski, Partial Differential Equations of Second Order, Monografie Matematyczne, vol. 53, Polish Scientific Publishers, Warsaw, 1971.

    MATH  Google Scholar 

  12. G. E. Latta, Singular Perturbation Problems, Doctoral Dissertation, Calif. Inst. of Tech., Pasadena, 1951.

    Google Scholar 

  13. N. Levinson, The First Boundary Value Problem for εΔu+A(x,y)ux+B(x,y)uy+C(x,y)u=D(x,y) for Small ε, Ann. Math. 51(1950), 428–445.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. J. L. Lions, Perturbation Singulieres dans les Problemes aux Limites et en Controle Optimal, Lecture Notes in Math., vol. 323, Springer Verlag, Berlin and New York, 1973.

    CrossRef  MATH  Google Scholar 

  15. J. Mauss, Etude des Solutions Asymptotiques de Problemes aux Limites Elliptiques pour des Domaines non Bornes, Compte Rendus Acad. Sci., Ser. A 269(1969), 25–28.

    MathSciNet  MATH  Google Scholar 

  16. N. Meyers and J. Serrin, The Exterior Dirichlet Problem for Second Order Elliptic Partial Differential Equations, J. Math. Mech. 9(1960), 513–538.

    MathSciNet  MATH  Google Scholar 

  17. R. E. O'Malley, Jr., Topics in Singular Perturbations, Adv. in Math. 2(1968), 365–470.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Akademische Verlagsgesellschaft M.B.H., Leipzig, 1927.

    MATH  Google Scholar 

  19. L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Proc. Third Int'l. Math. Congress Heidelberg 1904, Teubner, Leipzig, 1905, pp. 484–494; translation in NACA Memo. 452, 1928.

    MATH  Google Scholar 

  20. A. van Harten, Singularly Perturbed Non-Linear Second Order Elliptic Boundary Value Problems, Doctoral Thesis, Univ. of Utrecht, The Netherlands, 1975.

    Google Scholar 

  21. A. van Harten, Nonlinear Singular Perturbation Problems: Proofs of Correctness of a Formal Approximation Based on a Contraction Principle in a Banach Space, J. Math. Anal. Appl. 65(1978), 126–168.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. M. I. Vishik and L. A. Liusternik, Regular Degeneration and Boundary Layer for Linear Differential Equations with Small Parameter (in Russian), Uspekhi Mat. Nauk 12(1957), 3–122; translation in Amer. Math. Soc. Transl., Ser. 2 20(1961), 239–364.

    MathSciNet  Google Scholar 

  23. R. von Mises and K. O. Friedrichs, Fluid Dynamics, Springer Verlag, New York, 1971.

    CrossRef  MATH  Google Scholar 

  24. W. R. Wasow, Asymptotic Solution of Boundary Value Problems for the Differential Equation ΔU+λ(∂U/∂x)=λf(x,y), Duke Math. J. 11(1944), 405–415.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1982 Springer-Verlag

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Howes, F.A. (1982). The asymptotic solution of singularly perturbed Dirichlet problems with applications to the study of incompressible flows at high Reynolds number. In: Eckhaus, W., de Jager, E.M. (eds) Theory and Applications of Singular Perturbations. Lecture Notes in Mathematics, vol 942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094751

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  • DOI: https://doi.org/10.1007/BFb0094751

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