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.


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


Zbl 0996.82036
Full Text: DOI arXiv EuDML


[1] G. P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations , vol. 1. Linearized Steady Problems . Springer, New York, 1994. · Zbl 0949.35004
[2] D. Gilbarg and N. S. Trudinger. Differential Equations of Second Order , 2nd edition. Springer, Berlin, 1983. · Zbl 0562.35001
[3] O. A. Ladyzhenskaya and N. N. Uraltseva. Linear and Quasilinear Elliptic Equations . Nauka, Moscow, 1973 (English transl. of 1st edition as vol. 46 of the series Mathematics in Science and Engineering , Academic Press, New York, 1968). · Zbl 0164.13002
[4] F. Merkl and M. V. Wütrich. Infinite volume asymptotics of the ground state energy in scaled Poissonian potential. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 253-284. · Zbl 0996.82036
[5] V. V. Petrov. Sums of Independent Random Variables . Springer, New York, 1975. · Zbl 0322.60043
[6] A.-S. Sznitman. Capacity and principal eigenvalues: the method of enlargement of obstacles revisited. Ann. Probab. 25 (1997) 1180-1209. · Zbl 0885.60063
[7] A.-S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998. · Zbl 0973.60003
[8] R. Temam. Navier-Stokes Equations: Theory and Numerical Analysis . North-Holland, Amsterdam, 1979. · Zbl 0426.35003
[9] V. V. Yurinsky. Spectrum bottom and largest vacuity. Probab. Theory Related Fields 114 (1999) 151-175. · Zbl 0932.60100
[10] V. V. Yurinsky. Localization of spectrum bottom for the Stokes operator in a random porous medium. Siber. Math. J. 42 (2001) 386-413. · Zbl 0970.35102
[11] V. V. Yurinsky. On the smallest eigenvalue of the Stokes operator in a domain with fine-grained random boundary. Sibirsk. Mat. Zh. 47 (2006) 1414-1427. (Review in process) (English transl. in Siber. Math. J. 47 (2006) 1167-1178). · Zbl 1150.60033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.