  In this paper, we study the Landau equation under the Navier-Stokes scaling
in the torus for hard and moderately soft potentials. More precisely, we
investigate the Cauchy theory in a perturbative framework and establish some
new short time regularization estimates for our rescaled nonlinear Landau
equation. These estimates are quantified in time and optimal, indeed, we obtain
the instantaneous expected anisotropic gain of regularity (see [53] for the
corresponding hypoelliptic estimates on the linearized Landau collision
operator). Moreover, the estimates giving the gain of regularity in the
velocity variable are uniform in the Knudsen number. Intertwining these new
estimates on the Landau equation with estimates on the Navier-Stokes-Fourier
system, we are then able to obtain a result of strong convergence towards this
fluid system.
