On the Convergence from Boltzmann to Navier–Stokes–Fourier for General Initial Data

Abstract

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 the strategy from [M. Briant, S. Merino-Aceituno, and C. Mahout, Anal. Appl. (Singap.), 17 (2019), pp. 85–116] of writing the solution to the Boltzmann equation as the sum of a part with polynomial decay and a second one with Gaussian decay. The Gaussian part is treated with an approach reminiscent of the one from [I. Gallagher and I. Tristani, Ann. Henri Lebesgue, 3 (2020), pp. 561–614].