TY - JOUR
T1 - Existence and uniqueness of stationary solutions for 3D Navier-Stokes system with small random Forcing via stochastic cascades
AU - Bakhtin, Yuri
N1 - Funding Information:
I would like to thank M. D. Arnold, E. I. Dinaburg, J. Hanke, G. M. Molchan and Ya. G. Sinai for their useful comments. I am also grateful to anonymous referees for several helpful suggestions and for letting me know about (5). The author was partially supported by Russian Science Support Foundation and by Grant MK-2475.2004.1 of the President of the Russian Federation.
Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.
PY - 2006/1
Y1 - 2006/1
N2 - We consider the 3D Navier-Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following "one force-one solution" principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.
AB - We consider the 3D Navier-Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following "one force-one solution" principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.
KW - "One force-one solution" Principle
KW - Navier-Stokes system
KW - Stationary solution
KW - Stochastic cascades
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U2 - 10.1007/s10955-005-8014-x
DO - 10.1007/s10955-005-8014-x
M3 - Article
AN - SCOPUS:33645927641
SN - 0022-4715
VL - 122
SP - 351
EP - 360
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 2
ER -