Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation

Jingrui Wang; Keyan Wang

doi:10.3934/dcds.2017215

Article Contents

2017, Volume 37, Issue 9: 5003-5019. Doi: 10.3934/dcds.2017215

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Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation

Jingrui Wang^1, , and
Keyan Wang^2,

1.
School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China
2.
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

* Corresponding author: Jingrui Wang
* Corresponding author: Jingrui Wang

Received: May 31, 2016

Revised: March 31, 2017

Published: October 2017

The authors were in part supported in part by NSFC (Grant No. 11301338) and Shanghai Scientific Research Innovation Project (15ZZ098)

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Abstract

In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-α}(\mathbb{R}^{3})$ or $\dot{H}^{-α}(\mathbb{T}^{3})$ with $0<α≤ 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t≥0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlović and Staffilani on (SIMA) in which $α$ is restricted to $0<α<\frac{1}{4}$.

Keywords:

Navier-Stokes equation,
random initial data.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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