A lower bound for the principal eigenvalue of the Stokes operator in a random domain
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 1, p. 1-18

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 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.

Cet article est dédié à l'étude de la localisation de la valeur propre principale (VPP) de l'opérateur de Stokes sous la condition de Dirichlet sur la frontière d'un grand domaine aléatoire qui modélise l'espace des pores d' un bloc cubique de matière poreuse dotée d'une microstructure désordonnée. Le résultat principal est une borne inférieure asymptotiquement déterministe pour la VPP de l'opérateur correspondant á l'écoulement d'un liquide peu compressible en présence d'un petit potentiel positif aléatoire. Les arguments sont fondés sur la méthode proposée par F. Merkl et M. V. Wütrich pour localiser la VPP de l'opérateur de Schrödinger dans une situation similaire. Des efforts supplémentaires sont nécessaires pour combattre les complications provenant de la réduction à la classe de champs vectoriels de divergence nulle de la famille des fonctions utilisées pour caractériser la VPP de l'opérateur de Stokes par une formule variationnelle.

DOI : https://doi.org/10.1214/07-AIHP136
Classification:  60H25,  82B44,  60K40,  35P15,  35Q30,  47B80,  76S05,  82D30
Keywords: Stokes flow, principal eigenvalue, random porous medium, chess-board structure, infinite volume asymptotics, scaled random potential
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     author = {Yurinsky, V. V.},
     title = {A lower bound for the principal eigenvalue of the Stokes operator in a random domain},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {1},
     year = {2008},
     pages = {1-18},
     doi = {10.1214/07-AIHP136},
     zbl = {1173.82333},
     mrnumber = {2451568},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2008__44_1_1_0}
}
Yurinsky, V. V. A lower bound for the principal eigenvalue of the Stokes operator in a random domain. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 1, pp. 1-18. doi : 10.1214/07-AIHP136. http://www.numdam.org/item/AIHPB_2008__44_1_1_0/

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