On the eigenfunctions of the Stokes operator in a plane layer with a periodicity condition along it

B. V. Pal'tsev

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

Abstract: In Rummler’s previous paper, formulas for the eigenfunctions of the Stokes operator were derived (in a rather concise form) in the case of a three-dimensional layer with a periodicity condition in orthogonal directions along the layer. In this paper, eigenfunctions and associated pressures are constructed and studied in a plane $n$-dimensional (specifically, two-dimensional) layer with a periodicity condition in orthogonal directions along the layer. A very simple and useful velocity representation in terms of the pressure gradient is used. As a result, the derivation of formulas is considerably simplified and reduced without applying cumbersome expressions and the eigenfunctions are expressed in terms of the associated pressures. Two-sided estimates are given, and the asymptotic behavior of nontrivial eigenvalues of the Stokes operator is analyzed.

Key words: Stokes operator, $n$-dimensional plane layer, periodicity condition in orthogonal directions along a layer, eigenfunctions and associated pressures, asymptotic behavior of series of eigenvalues.

UDC: 519.634

Received: 10.09.2013

DOI: 10.7868/S0044466914020112

 English version: Computational Mathematics and Mathematical Physics, 2014, 54:2, 303–314

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