Entropic convergence and the linearized limit for the Boltzmann equation with external force
Tina Mai
TL;DR
Problem addressed: extend entropic convergence and the linearized limit for the Boltzmann equation to include external forces. Approach: adopt the external-force framework of $F$ with $\operatorname{div}_v F=0$ and $F\cdot v=0$, linearize around the Maxwellian $M$, use DiPerna–Lions renormalized solutions and the entropy framework to prove entropic convergence of fluctuations to the linearized dynamics $\partial_t g + v\cdot \nabla_x g + F\cdot \nabla_v g + Lg = 0$, and establish strong $L^1$ convergence. Findings: for any $g^{in}\in L^2(M\,dx\,dv)$, the fluctuations $g_\varepsilon$ converge to $g$ in $C([0,\infty); w-L^1((1+|v|^2)M\,dx\,dv))$, are entropically convergent for $t>0$, and the scaled collision term converges to the linearized form $g' + g'_* - g - g_*$. Significance: validates the linearization in kinetic theory with physically meaningful external forces and provides a robust pathway for deriving hydrodynamic limits in Vlasov–Boltzmann–type systems under external forcing.
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 Arsénio 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.