  In 2001, H. Koch and D. Tataru proved the existence of global in time
solutions to the incompressible Navier-Stokes equations in ${\mathbb{R}}^d$ for
initial data small enough in $BMO^{-1}$. We show in this article that the Koch
and Tataru solution has higher regularity. As a consequence, we get a decay
estimate in time for any space derivative, and space analyticity of the
solution. Also as an application of our regularity theorem, we prove a
regularity result for self-similar solutions.
