On the regularity of the Navier-Stokes equations: A well-posedness result and open questions

We study the incompressible Navier-Stokes equations on $R^n imes R^+$ and prove existence and uniqueness of a solution $u$ in $R^n \times [0,T]$ with $$||u|| x_T := \sup\limits_{x_i i\leq T} \max \left\{t^{1/2} |u(x,t) | , \left( t^{-n/2} \int_{B_{\sqrt{t} } (x) } \int^{1}_{0} |u|^2 dy d\hbar \right)^{1/2} \right\} \leq 2 ||u_0||_{BMO^{-1}_{T}}$$ provided the solution $v$ to the heat equation with the same initial data $u_0$ satisfies $$||u_0||_{BMO^{-1}_{T}} :=||v||_{X_T} \leq \delta$$ This condition on the initial data is local in space and frequency. It extends Kato's wellposedness result in $L^n$ since $L^n \subset BMO^{-1}_\infty$.The function space $BMO^{-1}_\infty$ is closely related to $BMO$:$f \in BMO^{-1}_infty$ if there exists a vector field $V \in BMO^n $ with $f=\nabla \cdot V$. This is joint work with D. Tataru.