Existence and uniqueness of stationary solutions for 3D Navier-Stokes system with small random Forcing via stochastic cascades

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)351-360
Number of pages10
JournalJournal of Statistical Physics
Volume122
Issue number2
DOIs
StatePublished - Jan 2006

Fingerprint

Navier-Stokes System
uniqueness
Stationary Solutions
Forcing
Cascade
cascades
Existence and Uniqueness
uniqueness theorem
stochastic processes
Uniqueness Theorem
Stochastic Processes
Fourier transform
Class

Keywords

  • "One force-one solution" Principle
  • Navier-Stokes system
  • Stationary solution
  • Stochastic cascades

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

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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.

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