### 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 language | English (US) |
---|---|

Pages (from-to) | 351-360 |

Number of pages | 10 |

Journal | Journal of Statistical Physics |

Volume | 122 |

Issue number | 2 |

DOIs | |

State | Published - Jan 2006 |

### Fingerprint

### 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

**Existence and uniqueness of stationary solutions for 3D Navier-Stokes system with small random Forcing via stochastic cascades.** / Bakhtin, Yuri.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Existence and uniqueness of stationary solutions for 3D Navier-Stokes system with small random Forcing via stochastic cascades

AU - Bakhtin, Yuri

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

UR - http://www.scopus.com/inward/record.url?scp=33645927641&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645927641&partnerID=8YFLogxK

U2 - 10.1007/s10955-005-8014-x

DO - 10.1007/s10955-005-8014-x

M3 - Article

AN - SCOPUS:33645927641

VL - 122

SP - 351

EP - 360

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -