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The Ekman flow and related problems: spectral theory and numerical analysis. (English) Zbl 1058.34025

An eigenvalue problem arising in the study of the stability of the Ekman boundary layer is of the form \[ M\zeta = \lambda N \zeta, \quad \zeta =(y,z)^T,\quad \lambda \in \mathbb{C}, \] where \(M,N\) are matrices whose entries are differential expressions of order up to 4 on \([0,\infty),\) and boundary conditions at \(0\) and \(\infty.\) The authors investigate various aspects of the spectral analysis of the operator \(L=M-\lambda N,\) where \(M,N,L\) are defined on subspaces of Sobolev spaces which give precise meaning to the formal boundary conditions. The essential spectrum is defined to be the set \(\sigma_{\text{ess}}(M,N) = \{ \lambda \in \mathbb{C}: M-\lambda N \text{ is not Fredholm}\}\) and in one of the main results of the paper the essential spectrum is located. There follows a detailed analysis of \(L^2\) solutions of related first-order systems of differential equations and this leads to the development of a Titchmarsh-Weyl coefficient \(M(\lambda)\) and an analysis of the convergence of approximations to the spectrum arising from regular problems. The theory is illustrated by numerical results.

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34L05 General spectral theory of ordinary differential operators
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