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arXiv · 1610.09747

Almost Sure Existence of Global Weak Solutions to the 3D Incompressible Navier-Stokes Equation

Jingrui Wang·Keyan Wang
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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 $\dot{H}^{-\alpha}(\mathbb{R}^{3})$ or $\dot{H}^{-\alpha}(\mathbb{T}^{3})$ with $0<\alpha\leq 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t\geq0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlovi\'{c} and Staffilani on (SIMA, [1])in which $\alpha$ is restricted to $0<\alpha<\frac{1}{4}$.

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Jingrui Wang, Keyan Wang. 2016-10-31. Almost Sure Existence of Global Weak Solutions to the 3D Incompressible Navier-Stokes Equation. https://arxiv.org/abs/1610.09747

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