Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades |
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Authors: | Yuri Bakhtin |
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Affiliation: | (1) Laboratory for Nonlinear Dynamics, International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow, Russia;(2) Mathematics Department, Duke University, Durham, NC, USA |
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Abstract: | 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. |
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Keywords: | Navier– Stokes system Stationary solution Stochastic cascades “ One force– one solution” Principle |
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