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

Abstract

We study the existence and uniqueness of the strong solution for the incompressible Navier–Stokes equations with the L 2 L^2 initial data and the periodic space domain T 3 \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 L^2 . Furthermore, if ‖ u 0 ‖ L 2 \|u_0\|_{L^2} is small, we show that the probability for the global existence and uniqueness of the solution is large.