We consider the Navier-Stokes system on R2. It is well-known that solutions with L2 data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in L∞Hσ(R2) and L2tHσ +1(R2), when 0 ≤ σ ≥ 1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in L2(R2). The question for the Lipschitzness of the map in. Hσ(R2), σ ≤ 1 remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form (φ,v,w)→∫R2(θφ/θx.θv/ θyθφθy.θvθx)wdx.
