Hydrodynamic limit for the non-cutoff Boltzmann equation

Abstract

This work deals with the non-cutoff Boltzmann equation for all types 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 Boltzmann equation towards the incompressible Navier–Stokes–Fourier system.