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

  • * Corresponding author: Jingrui Wang

    * Corresponding author: Jingrui Wang 
The authors were in part supported in part by NSFC (Grant No. 11301338) and Shanghai Scientific Research Innovation Project (15ZZ098)
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  • 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 $0<α<\frac{1}{4}$.

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


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