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Long time behavior for collisional strongly magnetized plasma in three space dimensions

Mihaï Bostan, Anh-Tuan Vu

Abstract

We consider the long time evolution of a population of charged particles, under strong magnetic fields and collision mechanisms. We derive a fluid model and justify the asymptotic behavior toward smooth solutions of this regime. In three space dimensions, a constraint ocurs along the parallel direction. For eliminating the corresponding Lagrange multiplier, we average along the magnetic lines.

Long time behavior for collisional strongly magnetized plasma in three space dimensions

Abstract

We consider the long time evolution of a population of charged particles, under strong magnetic fields and collision mechanisms. We derive a fluid model and justify the asymptotic behavior toward smooth solutions of this regime. In three space dimensions, a constraint ocurs along the parallel direction. For eliminating the corresponding Lagrange multiplier, we average along the magnetic lines.
Paper Structure (17 sections, 29 theorems, 300 equations)

This paper contains 17 sections, 29 theorems, 300 equations.

Table of Contents

  1. Keywords:
  2. AMS classification:
  3. Introduction
  4. Setting of the problem and main results
  5. The physical model and Scaling
  6. Main results
  7. Preliminaires
  8. Formal derivation of the limit model
  9. Convergence result
  10. Reformulation of the limit model
  11. A commutation formula for angular vector fields
  12. A commutation formula
  13. Cylindrical case
  14. Example of regular solutions for limit model
  15. Fundamental solution of Laplace's equation on ${\bf R}^2\times{\bf T}^1$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $B$ be a smooth magnetic field, such that $\inf_{x\in{\bf R}^3}B(x)=B_0 >0$. Assume that the initial particle densities $(f^ \varepsilon_{\mathrm{in}})_{ \varepsilon>0}$ satisfy $f^ \varepsilon_{\mathrm{in}}\geq 0$, $M_{\mathrm{in}}:=\sup_{ \varepsilon>0}M^ \varepsilon_{\mathrm{in}}<+\infty$, $U Let $T>0$. We denote by $f^ \varepsilon$ the weak solutions of equ:VPFP-Scale, equ:PoissonEpsi, and

Theorems & Definitions (33)

  • Theorem 2.1
  • Remark 2.1
  • Definition 3.1
  • Theorem 3.1
  • Lemma 3.1
  • Remark 4.1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 5.1
  • Proposition 6.1
  • ...and 23 more