  This work deals with the non-cutoff Boltzmann equation for all type of
potentials, in both the torus $\mathbf{T}^3$ and in the whole space
$\mathbf{R}^3$, under the incompressible Navier-Stokes scaling. We first
establish the well-posedness and decay of global mild solutions to this
rescaled Boltzmann equation in a perturbative framework, that is for solutions
close to the Maxwellian, obtaining in particular integrated-in-time
regularization estimates. We then combine these estimates with spectral-type
estimates in order to obtain the strong convergence of solutions to the
non-cutoff Boltzmannn equation towards the incompressible Navier-Stokes-Fourier
system.
