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

TL;DR

This paper establishes a deterministic lower bound for the principal eigenvalue of the Stokes operator in large random porous domains, extending localization techniques from Schrödinger operators to fluid dynamics.

Contribution

It introduces a novel lower bound for the Stokes principal eigenvalue in random domains, adapting localization methods to divergence-free vector fields.

Findings

Derived an asymptotically deterministic lower bound for the eigenvalue

Extended localization techniques to the Stokes operator in random media

Provided bounds applicable to porous materials with disordered micro-structure

Abstract

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 for the Stokes PE. The arguments are based on the method proposed by F. Merkl and M. V. W\"{u}trich for localization of the PE of the Schr\"{o}dinger operator in a similar setting. Some additional work is needed to circumvent the complications arising from the restriction to divergence-free vector fields of the class of test functions in the variational characterization of the Stokes PE.