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

V. V. Yurinsky

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 1, page 1-18
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

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 44.1 (2008): 1-18. <http://eudml.org/doc/77962>.

@article{Yurinsky2008,
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. 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.},
author = {Yurinsky, V. V.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Stokes flow; principal eigenvalue; random porous medium; chess-board structure; infinite volume asymptotics; scaled random potential},
language = {eng},
number = {1},
pages = {1-18},
publisher = {Gauthier-Villars},
title = {A lower bound for the principal eigenvalue of the Stokes operator in a random domain},
url = {http://eudml.org/doc/77962},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Yurinsky, V. V.
TI - A lower bound for the principal eigenvalue of the Stokes operator in a random domain
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 1
SP - 1
EP - 18
AB - 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.
LA - eng
KW - Stokes flow; principal eigenvalue; random porous medium; chess-board structure; infinite volume asymptotics; scaled random potential
UR - http://eudml.org/doc/77962
ER -

References

top
  1. G. P. Galdi. An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. 1. Linearized Steady Problems. Springer, New York, 1994. Zbl0949.35004MR1284205
  2. D. Gilbarg and N. S. Trudinger. Differential Equations of Second Order, 2nd edition. Springer, Berlin, 1983. Zbl0562.35001MR737190
  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). Zbl0164.13002MR244627
  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. Zbl0996.82036MR1899454
  5. V. V. Petrov. Sums of Independent Random Variables. Springer, New York, 1975. Zbl0322.60042MR388499
  6. A.-S. Sznitman. Capacity and principal eigenvalues: the method of enlargement of obstacles revisited. Ann. Probab. 25 (1997) 1180–1209. Zbl0885.60063MR1457616
  7. A.-S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998. Zbl0973.60003MR1717054
  8. R. Temam. Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam, 1979. Zbl0426.35003MR603444
  9. V. V. Yurinsky. Spectrum bottom and largest vacuity. Probab. Theory Related Fields 114 (1999) 151–175. Zbl0932.60100MR1701518
  10. V. V. Yurinsky. Localization of spectrum bottom for the Stokes operator in a random porous medium. Siber. Math. J. 42 (2001) 386–413. Zbl0970.35102MR1833169
  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). Zbl1150.60033MR2302853

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.