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From relativistic Vlasov-Maxwell to electron-MHD in the quasineutral regime

Antoine Gagnebin, Mikaela Iacobelli, Alexandre Rege, Stefano Rossi

TL;DR

This work rigorously justifies the quasineutral limit for the relativistic Vlasov–Maxwell system under analytic regularity, deriving the kinetic electron-MHD (e-MHD) model as the limit when the Debye-scale parameter $\varepsilon\to0$. The authors construct local-in-time solutions to the Euler–Maxwell system $(\mathrm{EM}^\varepsilon)$ with uniform bounds in analytic norms and, crucially, identify and subtract fast electromagnetic oscillations via dispersive correctors. The limiting system is shown to be well-posed in the analytic framework, with strong convergence of the filtered momentum and fields to the e-MHD dynamics, together with a detailed description of oscillatory correctors $d_{0,\pm},d_{1,\pm},d_{2,\pm}$. This result provides a rigorous underpinning for e-MHD reductions used in tokamak and stellarator plasma modeling and offers a framework for handling electromagnetic oscillations in singular limits. The methodology hinges on a Grenier-style multifluid decomposition, Helmholtz–Hodge splitting of the electric field, and precise dispersive estimates, opening avenues for extending the analysis to broader kinetic-electromagnetic limits and dispersive regimes.

Abstract

We study the quasineutral limit for the relativistic Vlasov-Maxwell system in the framework of analytic regularity. Following the high regularity approach introduced by Grenier [44] for the Vlasov-Poisson system, we construct local-in-time solutions with analytic bounds uniform in the quasineutrality parameter $\varepsilon$. In contrast to the electrostatic case, the presence of a magnetic field and a solenoidal electric component leads to new oscillatory effects that require a refined decomposition of the electromagnetic fields and the introduction of dispersive correctors. We show that, after appropriate filtering, solutions converge strongly as $\varepsilon$ tends to zero to a limiting system describing kinetic electron magnetohydrodynamics (e-MHD). This is the first strong convergence result for the Vlasov-Maxwell system in the quasineutral limit under analytic regularity assumptions, providing a rigorous justification for the e-MHD reduction, widely used in modelling plasmas in tokamaks and stellarators.

From relativistic Vlasov-Maxwell to electron-MHD in the quasineutral regime

TL;DR

This work rigorously justifies the quasineutral limit for the relativistic Vlasov–Maxwell system under analytic regularity, deriving the kinetic electron-MHD (e-MHD) model as the limit when the Debye-scale parameter . The authors construct local-in-time solutions to the Euler–Maxwell system with uniform bounds in analytic norms and, crucially, identify and subtract fast electromagnetic oscillations via dispersive correctors. The limiting system is shown to be well-posed in the analytic framework, with strong convergence of the filtered momentum and fields to the e-MHD dynamics, together with a detailed description of oscillatory correctors . This result provides a rigorous underpinning for e-MHD reductions used in tokamak and stellarator plasma modeling and offers a framework for handling electromagnetic oscillations in singular limits. The methodology hinges on a Grenier-style multifluid decomposition, Helmholtz–Hodge splitting of the electric field, and precise dispersive estimates, opening avenues for extending the analysis to broader kinetic-electromagnetic limits and dispersive regimes.

Abstract

We study the quasineutral limit for the relativistic Vlasov-Maxwell system in the framework of analytic regularity. Following the high regularity approach introduced by Grenier [44] for the Vlasov-Poisson system, we construct local-in-time solutions with analytic bounds uniform in the quasineutrality parameter . In contrast to the electrostatic case, the presence of a magnetic field and a solenoidal electric component leads to new oscillatory effects that require a refined decomposition of the electromagnetic fields and the introduction of dispersive correctors. We show that, after appropriate filtering, solutions converge strongly as tends to zero to a limiting system describing kinetic electron magnetohydrodynamics (e-MHD). This is the first strong convergence result for the Vlasov-Maxwell system in the quasineutral limit under analytic regularity assumptions, providing a rigorous justification for the e-MHD reduction, widely used in modelling plasmas in tokamaks and stellarators.
Paper Structure (22 sections, 11 theorems, 536 equations)

This paper contains 22 sections, 11 theorems, 536 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Notation
  4. Main results
  5. Discussion of proofs
  6. Analysis of fast electromagnetic oscillations
  7. Plan of the paper
  8. Technical inequalities
  9. Local-in-time (uniform in Lg) solutions to the Euler--Maxwell system
  10. A priori estimates on Lg
  11. A priori estimates on Lg
  12. Estimates on Lg and Lg
  13. Estimates on Lg and Lg
  14. Estimates on Lg
  15. Iterative scheme
  16. ...and 7 more sections

Key Result

Theorem 1.1

(Local-in-time (uniform in $\varepsilon$) solutions to (EM$^\varepsilon$) system sys:EM.) Given $\varepsilon>0$ and a probability space $(M,\mu)$, let $\{\rho^\varepsilon_\Theta(0), \xi^\varepsilon_\Theta(0)\}_{\Theta \in M}$ a bounded family belonging to $\widetilde{B}_{\delta_0}\times \widetilde{B Moreover, assume that there exist $\delta_1>\delta_0$ and $C_0>0$ such that Then there exist $\var

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 14 more