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The Behavior of Solutions of a Nonlinear Boundary Layer Equation

  • Chunqing Lu
  • William C. Troy
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 13)

Abstract

We investigate the behavior of solutions of the equation φ′″ +1 -φ + λ(φφ″ - (φ)′2) = 0. This equation arises in modelling large scale ocean circulation with particular emphasis on the behavior of the Gulf Stream. There are two sets of physically interesting boundary conditions, namely φ(0) = φ′(0) = 0 and φ(∞) = 1 (no-slip conditions) or φ(0) = φ′(0) = 0 and φ(∞) = 1 (stress-free conditions). For each of these problems we prove that solutions exist if |λ| is small. However, if λ ≤ -9 then there is no solution of the no-slip problem. If λ ≤ -(2)1/3 then the stress-free problem has no solution.

Keywords

Banach Space Gulf Stream Maximal Interval Vorticity Equation Nonlinear Diffusion Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    G.R. Ierley and O.G. Ruehr, Analytic and numerical solutions of a nonlinear boundary-layer problem, Studies in Applied Math. 75 (1986), 1–36.MATHMathSciNetGoogle Scholar
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    J.B. McLeod, The existence of axially symmetric flow above a rotating disk, Proc. Roy. Soc. London A 324 (1971), 391–414.CrossRefMATHMathSciNetGoogle Scholar
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    J. Smoller, “Shock waves and reaction-diffusion equations,” Grundlehren der Math, Wissenschaften, Vol. 258, Springer-Verlag, New York, 1983.Google Scholar
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    W.C. Troy, Solutions of a nonlinear boundary layer problem arising in physical oceanography, SIAM J. Math. Anal, (submitted).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Chunqing Lu
    • 1
  • William C. Troy
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of PittsburghPittsburghUSA

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