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 $$||u||x_T:=\underset{x_{i}i\le T}{sup}max\{t^{1/2}|u(x,t)|,{\left(t^{-n/2}{\int }_{B_{\sqrt{t}}(x)}{\int }_0^1|u{|}^{2}dyd\hslash \right)}^{1/2}\}\le 2||u_0|{|}_{BM{O}_T^{-1}}$$ provided the solution $v$ to the heat equation with the same initial data $u_0$ satisfies $$||u_0|{|}_{BM{O}_T^{-1}}:=||v|{|}_{X_T}\le \delta$$ This condition on the initial data is local in space and frequency. It extends Kato's wellposedness result in $L^n$ since $L^n\subset BM{O}_{\mathrm{\infty }}^{-1}$.The function space $BM{O}_{\mathrm{\infty }}^{-1}$ is closely related to $BMO$:$f\in BM{O}_i^{-1}nfty$ if there exists a vector field $V\in BM{O}^n$ with $f=\mathrm{\nabla }\cdot V$. This is joint work with D. Tataru.