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We study the Boltzmann equation on the $d$-dimensional torus in aperturbative setting around a global equilibrium under the Navier-Stokeslinearisation. We use a recent functional analysis breakthrough to prove thatthe linear part of the equation generates a $C^0$-semigroup with exponentialdecay in Lebesgue and Sobolev spaces with polynomial weight, independently onthe Knudsen number. Finally we show a Cauchy theory and an exponential decayfor the perturbed Boltzmann equation, uniformly in the Knudsen number, inSobolev spaces with polynomial weight. The polynomial weight is almost optimaland furthermore, this result only requires derivatives in the space variableand allows to connect to solutions to the incompressible Navier-Stokesequations in these spaces.

From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight