  In this work, we prove the convergence of strong solutions of the Boltzman
equation, for initial data having polynomial decay in the velocity variable,
towards those of the incompressible Navier-Stokes-Fourier system. We show in
particular that the solutions of the rescaled Boltzmann equation do not blow up
before their hydrodynamic limit does. This is made possible by adapting a
strategy introduced by M. Briant, S. Merino and C. Mouhot of writing the
solution to the Boltzmann equation as the sum a part with polynomial decay and
a second one with Gaussian decay. The Gaussian part is treated with an approach
reminiscent of the one used by I. Gallagher and I. Tristani.
