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Global regularity for solutions of the three dimensional Navier-Stokes equation with almost two dimensional initial data

In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier--Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates between the global existence of smooth solutions for the two dimensional Navier--Stokes equation with arbitrarily large initial data, and the global existence of smooth solutions for the Navier--Stokes equation in three dimensions with small initial data in $\dot{H}^\frac{1}{2}$. This result states that the closer the initial data is to being two dimensional, the larger the initial data can be in $\dot{H}^\frac{1}{2}$ while still guaranteeing the global existence of smooth solutions. In the whole space, this set of almost two dimensional initial data is unbounded in the critical space $\dot{H}^\frac{1}{2},$ but is bounded in the critical Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$ for all $2<p\leq +\infty.$ On the torus, however, this approach does give examples of arbitrarily large initial data in the endpoint Besov space $\dot{B}^{-1}_{\infty,\infty}$ that generate global smooth solutions to the Navier--Stokes equation. In addition to these new results, we will also sharpen the constants in a number of previously known estimates for the growth of solutions to the Navier--Stokes equation and clarify the relationship between certain component reduction type regularity criteria.