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The Euler--Maxwell system for electrons: global solutions in $2D$

Yu Deng, Alexandru D. Ionescu, Benoit Pausader

TL;DR

The work proves global stability for the 2D one-fluid Euler–Maxwell electrons system near a constant neutral background by marrying high-order energy methods with a delicate dispersive Fourier analysis tailored to space–time resonances. The authors introduce a quasilinear reformulation and a Z-norm capturing resonant outputs, then decompose nonlinearities into nonresonant, strongly semilinear, and resonant parts, deriving sharp L^2 and L^∞ bounds. A key innovation is a restricted nondegeneracy property for the time-resonant set, enabling a novel L^2 bound on localized Fourier integral operators and a partial normal form that controls the strongest modulations. Together, these ingredients yield almost optimal pointwise decay $t^{-1+\kappa}$ and establish global existence and scattering-type behavior for small, localized data, contributing a robust framework for quasilinear dispersive systems with slow decay and large resonance sets.

Abstract

A basic model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. In this paper we consider the "one-fluid" Euler--Maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background.

The Euler--Maxwell system for electrons: global solutions in $2D$

TL;DR

The work proves global stability for the 2D one-fluid Euler–Maxwell electrons system near a constant neutral background by marrying high-order energy methods with a delicate dispersive Fourier analysis tailored to space–time resonances. The authors introduce a quasilinear reformulation and a Z-norm capturing resonant outputs, then decompose nonlinearities into nonresonant, strongly semilinear, and resonant parts, deriving sharp L^2 and L^∞ bounds. A key innovation is a restricted nondegeneracy property for the time-resonant set, enabling a novel L^2 bound on localized Fourier integral operators and a partial normal form that controls the strongest modulations. Together, these ingredients yield almost optimal pointwise decay and establish global existence and scattering-type behavior for small, localized data, contributing a robust framework for quasilinear dispersive systems with slow decay and large resonance sets.

Abstract

A basic model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. In this paper we consider the "one-fluid" Euler--Maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background.

Paper Structure

This paper contains 39 sections, 44 theorems, 657 equations.

Table of Contents

  1. Introduction
  2. The one-fluid Euler--Maxwell system in 2D and the Main Theorem
  3. Challenges and main ingredients
  4. Energy estimates
  5. Dispersive analysis
  6. Organization of the paper
  7. Functions spaces and the main proposition
  8. Notation, atomic decomposition, and the $Z$-norm
  9. Littlewood-Paley projections
  10. Quadratic phases, linear profiles and norms
  11. The main bootstrap proposition
  12. Main lemmas
  13. Operator Estimates
  14. Weyl paradifferential calculus
  15. Integration by parts
  16. ...and 24 more sections

Key Result

Theorem 1.1

Assume that $\overline{c}>0$, $h:(-\overline{c},\overline{c})\to\mathbb{R}$ is an analytic function, and $d=h'(0)\in(0,1)$. Assume that $(\rho_0,u_0,\widetilde{E}_0,\widetilde{b}_0):\mathbb{R}^2\to\mathbb{R}\times\mathbb{R}^2\times\mathbb{R}^2\times\mathbb{R}$ is small, smooth, and localized data th and the smallness condition Here $\overline{\epsilon}>0$ is sufficiently small, $N_{0},N_1$ are su

Theorems & Definitions (74)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 64 more