The Behavior of Solutions of a Nonlinear Boundary Layer Equation
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.
KeywordsBanach Space Gulf Stream Maximal Interval Vorticity Equation Nonlinear Diffusion Equation
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- 3.J. Smoller, “Shock waves and reaction-diffusion equations,” Grundlehren der Math, Wissenschaften, Vol. 258, Springer-Verlag, New York, 1983.Google Scholar
- 4.W.C. Troy, Solutions of a nonlinear boundary layer problem arising in physical oceanography, SIAM J. Math. Anal, (submitted).Google Scholar