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

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 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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