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A lower bound for the principal eigenvalue of the Stokes operator in a random domain

V. V. Yurinsky — 2008

Annales de l'I.H.P. Probabilités et statistiques

This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...

Weak solutions for a well-posed Hele-Shaw problem

S. N. AntontsevA. M. MeirmanovV. V. Yurinsky — 2004

Bollettino dell'Unione Matematica Italiana

We analyze existence and uniqueness of weak solutions to the well-posed Hele-Shaw problem under general conditions on the fixed boundaries and non-homogeneous governing equation in the unknown domain and non-homogeneous dynamic condition on the free boundary. Our approach allows us also to minimize the restrictions on the boundary and initial data. We derive several estimates on the solutions in B V spaces, prove a comparison theorem, and show that the solution depends continuously on the initial...

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