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A lower bound for the principal eigenvalue of the Stokes operator in a random domain. (English) Zbl 1173.82333

Summary: 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. Wütrich [Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 3, 253–284 (2002; Zbl 0996.82036)] for localization of the PE of the Schrödinger 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.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60K40 Other physical applications of random processes
60H25 Random operators and equations (aspects of stochastic analysis)
35P15 Estimates of eigenvalues in context of PDEs
35Q30 Navier-Stokes equations
47B80 Random linear operators
76S05 Flows in porous media; filtration; seepage
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
76D07 Stokes and related (Oseen, etc.) flows

Citations:

Zbl 0996.82036
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References:

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