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We study the Boltzmann equation with external forces, not necessarilyderiving from a potential, in the incompressible Navier-Stokes perturbativeregime. On the torus, we establish local-in-time, for any time, Cauchy theoriesthat are independent of the Knudsen number in Sobolev spaces. The existence isproved around a time-dependent Maxwellian that behaves like the globalequilibrium both as time grows and as the Knudsen number decreases. We combinehypocoercive properties of linearized Boltzmann operators with linearizationaround a time-dependent Maxwellian that catches the fluctuations of thecharacteristics trajectories due to the presence of the force. This uniformtheory is sufficiently robust to derive the incompressibleNavier-Stokes-Fourier system with an external force from the Boltzmannequation. Neither smallness, nor time-decaying assumption is required for theexternal force, nor a gradient form, and we deal with general hard potentialand cut-off Boltzmann kernels. As a by-product the latest general theories forunit Knudsen number when the force is sufficiently small and decays in time arerecovered.

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