  We study the Boltzmann equation on the $d$-dimensional torus in a
perturbative setting around a global equilibrium under the Navier-Stokes
linearisation. We use a recent functional analysis breakthrough to prove that
the linear part of the equation generates a $C^0$-semigroup with exponential
decay in Lebesgue and Sobolev spaces with polynomial weight, independently on
the Knudsen number. Finally we show a Cauchy theory and an exponential decay
for the perturbed Boltzmann equation, uniformly in the Knudsen number, in
Sobolev spaces with polynomial weight. The polynomial weight is almost optimal
and furthermore, this result only requires derivatives in the space variable
and allows to connect to solutions to the incompressible Navier-Stokes
equations in these spaces.
