Entropic convergence and the linearized limit for the Boltzmann equation with external force

TL;DR

This paper extends entropic convergence results for the Boltzmann equation to include external forces, establishing conditions under which solutions converge strongly to the linearized limit, even with significant external forces involved.

Contribution

It generalizes previous entropic convergence results to the Boltzmann equation with external force, providing rigorous conditions and proofs for the linearized limit in this context.

Findings

Solutions with external force converge entropically to the linearized Boltzmann solution.

The convergence is strong in L^1 for fluctuations around Maxwellian equilibrium.

New conditions on the external force ensure the validity of the convergence.

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.