On the Lipschitzness of the solution map for the 2 D Navier-Stokes system

We consider the Navier-Stokes system on R². It is well-known that solutions with $L^2$ data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in $L^\infty $H^σ (R²) and $L^2_t$H^σ+1 (R²), when $0\leq $ σ<1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in $L^2$(R²). The question for the Lipschitzness of the map in H^σ (R²), σ$\geq 1$ remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form $(\phi, v, w)\to \int$_R²(∂Φ/∂x ∂v/∂y - ∂Φ/∂y ∂v/∂x)wdx.