Forces for the Navier-Stokes equations and the Koch and Tataru theorem

We consider the Cauchy problem for the incompressible Navier--Stokes equations on the whole space \(\mathbb{R}^3\), with initial value \(\vec u_0\in {\rm BMO}^{-1}\) (as in Koch and Tataru's theorem) and with force \(\vec f=\Div \mathbb{F}\) where smallness of \(\mathbb{F}\) ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin's solutions (in \(L^2\mathcal{F}^{-1}L^1\)) or solutions in the multiplier space \(\mathcal{M}(\dot H^{1/2,1}_{t,x}\mapsto L^2_{t,x})\).