Random data Cauchy theory for the incompressible three dimensional Navier–Stokes equations

Zhang, Ting; Fang, Daoyuan

doi:10.1090/S0002-9939-2011-10762-7

Random data Cauchy theory for the incompressible three dimensional Navier–Stokes equations
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by Ting Zhang and Daoyuan Fang PDF

Proc. Amer. Math. Soc. 139 (2011), 2827-2837 Request permission

Abstract:

We study the existence and uniqueness of the strong solution for the incompressible Navier–Stokes equations with the $L^2$ initial data and the periodic space domain $\mathbb {T}^3$. After a suitable randomization, we are able to construct the local unique strong solution for a large set of initial data in $L^2$. Furthermore, if $\|u_0\|_{L^2}$ is small, we show that the probability for the global existence and uniqueness of the solution is large.

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