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The Boltzmann equation with an external force on the torus: incompressible Navier–Stokes–Fourier hydrodynamical limit

Marc Briant, Arnaud Debussche and Julien Vovelle

Vol. 4 (2022), No. 4, 597–628
Abstract

We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier–Stokes perturbative regime. On the torus, we establish Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of the characteristics trajectories due to the presence of the force. This uniform theory is sufficiently robust to derive the incompressible Navier–Stokes–Fourier system with an external force from the Boltzmann equation. Neither smallness nor a time-decaying assumption is required for the external force, nor a gradient form, and we deal with general hard potential and cut-off Boltzmann kernels. As a by-product, the latest general theories for unit Knudsen number when the force is sufficiently small and decays in time are recovered.

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Keywords
Boltzmann equation with external force, hydrodynamical limit, incompressible Navier–Stokes equation, hypocoercivity, Knudsen number
Mathematical Subject Classification
Primary: 35Q20, 82C40
Secondary: 76P05
Milestones
Received: 2 October 2020
Revised: 7 December 2021
Accepted: 16 May 2022
Published: 21 January 2023
Authors
Marc Briant
Université de Paris
MAP5, CNRS, UMR 8145
Paris
France
Arnaud Debussche
ENS Rennes
Bruz
France
Julien Vovelle
UMPA, UMR 5669, CNRS
ENS de Lyon site Monod
Lyon
France