# 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 *F _{t}* contained in a large cubic block [-

*t*,

*t*]

^{d},

*t*→∞. In individual unit cubic cells, the complements to the flow domain are

*S*= (

_{z}*z*+[-1/2, 1/2]

^{d})

*F*for each

_{t}*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 [2] 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 [4] for planar flows to dimensions

*d*>2. The method of [2] is used with later modifications [3].

- Sznitman A.-S.,
*Brownian Motion, Obstacles and Random Media*, (Springer-Verlag, Berlin-Heidelberg-New York, 1998). - Merkl F., Wütrich N. V., "Infinite volume asymptotics of the ground state energy in scaled Poissonian potential",
*Ann.Inst. H. Poincaré - PR***38**(3) (2002), 253-284. - Yurinsky V. V., "On the principal eigenvalue of the Schrödinger operator with a scaled random potential", in
*Contemporary Mathematics*(2003) vol. 339, 201-216. - Yurinsky V. V., "Localization of spectrum bottom for the Stokes operator in a random porous medium",
*Siber. Math. J.***42**(2) (2001), 386-413.