On principal eigenvalue of Stokes operator in large random domain
by Vadim Yurinsky (Universidade da Beira Interior)
This communication deals with localization of the principal eigenvalue (PE) of the Stokes operator under the Dirichlet condition on a random fine-grained boundary of the flow domain Ft contained in a large cubic block [-t,t]d, t→∞. In individual unit cubic cells, the complements to the flow domain are Sz = (z+[-1/2, 1/2]d) Ft for each z from the integer lattice. The random microstructure is assumed identically distributed and essentially independent in distinct cells. In this setting, the PE exhibits deterministic behavior as the volume of the containing block goes to infinity. For the Laplace and Schrödinger operators, problems of this kind have been studied since mid-eighties (see [1,2]). A major part of exposition deals with adaptation to the Stokes operator of the approach used in  to characterize large volume asymptotic behaviour of the PE of the Schrödinger operator. One of the results is an extension of the upper bound on the Stokes PE obtained in  for planar flows to dimensions d>2. The method of  is used with later modifications .
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