Localization of spectrum bottom for the Stokes operator in a random porous medium. (English. Russian original) Zbl 0970.35102

Yurinsky, V. V.

Sib. Math. J. 42, No. 2, 386-413 (2001); translation from Sib. Mat. Zh. 42, No. 2, 451-483 (2001).

The article is devoted to the spectral theory for the Stokes operator in a random porous medium. The author exposes a series of theorems which assert that the principal eigenvalue of the Stokes system \[ \Delta U - \nabla p + f = 0,\quad \nabla\cdot U = 0 \] on a random subdomain of a cube with edge \(r\) has the order \(O(\ln^{-2/d}r)\) with probability arbitrarily close to unity. A determinate interval is also indicated which contains this random variable with probability arbitrarily close to unity.

Reviewer: V.Grebenev (Novosibirsk)

MSC:

35Q30	Navier-Stokes equations
35P15	Estimates of eigenvalues in context of PDEs
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

Stokes operator; random porous medium; spectral theory; principal eigenvalue; localization of spectrum bottom