The second iterate for the Navier-Stokes equation

We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak strong uniqueness for Koch-Tataru solutions; and finally an instability result in $B^{-1}_{\infty,q}$, for q>2.