On massless electron limit for a multispecies kinetic system with external magnetic field

We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions confined by an external nonconstant magnetic field. Then we derive a kinetic-fluid model when the mass ratio $m_e/m_i$ tends to zero. Each species initially obeys a Vlasov-type equation and the electrostatic coupling follows from a Poisson equation. In our modeling, ions are assumed non-collisional while a Fokker-Planck collision operator is taken into account in the electron equation. As the mass ratio tends to zero we show convergence to a new system where the macroscopic electron density satisfies an anisotropic drift-diffusion equation. To achieve this task, we overcome some specific technical issues of our model such as the strong effect of the magnetic field on electrons and the lack of regularity at the limit. With methods usually adapted to diffusion limit of collisional kinetic equations and including renormalized solutions, relative entropy dissipation and velocity averages, we establish the rigorous derivation of the limit model.

💡 Research Summary

The paper addresses the rigorous derivation of a reduced kinetic‑fluid model for a two‑species plasma (electrons and ions) subjected to an external, non‑uniform magnetic field. Starting from the three‑dimensional Vlasov‑Poisson‑Fokker‑Planck system, the authors consider the asymptotic regime where the electron‑to‑ion mass ratio ε = mₑ/mᵢ tends to zero while keeping the Debye length, coupling parameter, and magnetic strength fixed (δ = η = µ = 1 in the nondimensionalization). In this scaling the electron equation contains dominant terms of order 1/ε: the Lorentz force (v ∧ B)·∇ᵥ and the Fokker‑Planck collision operator. Ions remain collisionless and evolve according to a Vlasov equation coupled to Poisson’s equation.

Because the electron distribution fₑ^ε is only known to belong to L¹_{x,v} uniformly in ε, the nonlinear term ∇ₓφ^ε·∇ᵥfₑ^ε cannot be interpreted directly. To overcome this, the authors adopt the DiPerna‑Lions framework of renormalized solutions. They introduce two families of renormalizations: (i) β‑renormalizations for smooth functions β satisfying growth conditions, yielding an equation for β(fₑ^ε); (ii) a weighted renormalization θ^ε_λ = fₑ^ε + λM, where M is the Maxwellian, leading to an auxiliary equation that captures entropy dissipation. These formulations allow the derivation of uniform (in ε) a priori bounds: mass conservation, energy estimates, and relative entropy dissipation.

The paper then establishes uniform estimates: fᵢ^ε is bounded in L^∞t(L¹{x,v}∩L^∞_{x,v}), fₑ^ε in L^∞t L¹{x,v}, and the electric field ∇ₓφ^ε in L^∞t L²_x. Crucially, the entropy inequality provides an ε‑independent bound on ∇ᵥ√{fₑ^ε+λM} in L²{t,x,v}, which together with the velocity‑averaging lemma yields compactness of the macroscopic electron density nₑ^ε and of the ion distribution.

A central technical difficulty is the strong magnetic term (v ∧ B)·∇ᵥfₑ^ε, which oscillates at frequency 1/ε. The authors exploit a precise cancellation between this term and the diffusion generated by the Fokker‑Planck operator. By writing the combined operator as L_A f = ∇ᵥ·(A(t,x)v f + ∇ᵥf) with A = I + v ∧ B, they show that A is invertible and its inverse A^{-1} is uniformly bounded because B is bounded. Consequently, the limit diffusion matrix D(t,x) = A^{-1}(t,x) is well defined and belongs to L^∞(0,T;L^∞_x).

Using the Aubin‑Lions lemma and velocity‑averaging results, the authors prove strong convergence (up to subsequences) of fᵢ^ε to a limit fᵢ and of nₑ^ε to nₑ. Passing to the limit ε→0 in the renormalized equations yields the following coupled system:

• Ion kinetic equation (unchanged): ∂_t fᵢ + v·∇ₓ fᵢ – ∇ₓφ·∇ᵥ fᵢ + (v ∧ B)·∇ᵥ fᵢ = 0.

• Electron fluid equation: ∂_t nₑ + ∇ₓ·jₑ = 0, jₑ = – D(t,x) (∇ₓ nₑ – nₑ ∇ₓ φ).

• Poisson equation: –Δₓ φ = nᵢ – nₑ, where nᵢ = ∫ fᵢ dv.

The diffusion matrix D(t,x) = A^{-1}(t,x) is explicitly given by \