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Abstract

This paper extends the results regarding entropic convergence and the strong linearized limit for the Boltzmann equation (without external force) in [C. David Levermore. Entropic convergence and the linearized limit for the Boltzmann equation. Communications in Partial Differential Equations, 18(7-8):1231--1248, 1993] to the case of the Boltzmann equation with external force. Our starting point is the Boltzmann equation with an external force introduced in [Diogo Ars\'enio and Laure Saint-Raymond. From the Vlasov--Maxwell--Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics, EMS Press, 2019], we then find new conditions on the force and rigorously prove the maintaining result by Levermore. More specifically, any sequence of DiPerna-Lions renormalized solutions of the Boltzmann equation with external force are shown to have fluctuations (about the global Maxwellian equilibrium $M$) that converge entropically (and hence strongly in $L^1$) to the solution of the linearized Boltzmann equation for any positive time, given that its initial fluctuations about $M$ converge entropically to the provided $L^2$ initial data of the linearized equation, where the force can be physically significant.