In this talk, I will discuss the construction of global in time critical mild solutions for the forced incompressible fractional Navier-Stokes equations, which differ from the classical Navier-Stokes equations through the introduction of a fractional diffusion. These solutions are obtained via a fixed point formulation which relies on suitable estimates for the non linearity, the initial velocity and the external force. Many functional spaces have been considered in the literature, but the focus of this talk will be on some Banach function spaces exploiting the pointwise decay of the kernel appearing in the nonlinearity.

After first recalling the general framework of mild solutions and a few main difficulties in the classical case, I will then discuss some functional spaces allowing us to construct solutions with initial data in the maximal critical Besov space only reachable in the fractional case. One key point of the discussion will be to highlight how despite the presence of a natural embedding of such spaces for the velocity, such structure is much less obvious for the choice of the external force.