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Ill-Posedness of the Incompressible Euler--Maxwell Equations in the Yudovich Class

Haroune Houamed

TL;DR

The paper proves that the 2D Euler–Maxwell system is mildly ill-posed in the Yudovich class when the Normal Structure condition on the electromagnetic field is violated, and this ill-posedness persists for any finite speed of light $c>0$. By expanding around a horizontal magnetic background, the Lorentz force is shown to contain a leading singular term $\mathcal{R}\omega$ and a lower-order remainder; the authors develop a Besov-space framework to control transport, Maxwell dynamics, and commutator effects, and they construct data sequences that inflate the vorticity in short time, breaking continuity of the solution map. The result sharpens the understanding of how electromagnetic coupling can disrupt Yudovich theory and highlights the central role of the Riesz-type mechanism in the ill-posedness. The findings are robust on both $\mathbb{R}^2$ and $\mathbb{T}^2$, and they connect to related ill-posedness phenomena in MHD and Euler–Riesz-type models, offering insights for future work on strong ill-posedness in plasma-fluid systems.

Abstract

It was shown recently by Arsénio and the author that the two-dimensional incompressible Euler--Maxwell system is globally well-posed in the Yudovich class, provided that the electromagnetic field enjoys appropriate conditions, including the Normal Structure. In this paper, we prove that this assumption is sharp, in the sense that the Euler--Maxwell system becomes ill-posed in the Yudovich class for initial data that do not obey the Normal Structure condition. The proof applies to both the whole plane and the two-dimensional torus, and holds for any value of the speed of light $c\in (0,\infty)$. This is achieved by expanding the magnetic field around a horizontal background and showing that the Lorentz force can be decomposed into two parts: the first is in the form of a singular operator acting on the vorticity, and the second, a "remainder", is of lower order when analyzed in a specific time regime.

Ill-Posedness of the Incompressible Euler--Maxwell Equations in the Yudovich Class

TL;DR

The paper proves that the 2D Euler–Maxwell system is mildly ill-posed in the Yudovich class when the Normal Structure condition on the electromagnetic field is violated, and this ill-posedness persists for any finite speed of light . By expanding around a horizontal magnetic background, the Lorentz force is shown to contain a leading singular term and a lower-order remainder; the authors develop a Besov-space framework to control transport, Maxwell dynamics, and commutator effects, and they construct data sequences that inflate the vorticity in short time, breaking continuity of the solution map. The result sharpens the understanding of how electromagnetic coupling can disrupt Yudovich theory and highlights the central role of the Riesz-type mechanism in the ill-posedness. The findings are robust on both and , and they connect to related ill-posedness phenomena in MHD and Euler–Riesz-type models, offering insights for future work on strong ill-posedness in plasma-fluid systems.

Abstract

It was shown recently by Arsénio and the author that the two-dimensional incompressible Euler--Maxwell system is globally well-posed in the Yudovich class, provided that the electromagnetic field enjoys appropriate conditions, including the Normal Structure. In this paper, we prove that this assumption is sharp, in the sense that the Euler--Maxwell system becomes ill-posed in the Yudovich class for initial data that do not obey the Normal Structure condition. The proof applies to both the whole plane and the two-dimensional torus, and holds for any value of the speed of light . This is achieved by expanding the magnetic field around a horizontal background and showing that the Lorentz force can be decomposed into two parts: the first is in the form of a singular operator acting on the vorticity, and the second, a "remainder", is of lower order when analyzed in a specific time regime.
Paper Structure (13 sections, 11 theorems, 86 equations)

This paper contains 13 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The model of interest
  3. Motivation and Main result
  4. Heuristic discussion
  5. Roadmap to ill-posedness in the Yudovich class
  6. Tool box
  7. Functional spaces and basic properties
  8. On Riesz operator and the flow map
  9. Transport equation in Besov spaces
  10. Maxwell equation in Besov spaces
  11. Local-in-time smooth solutions
  12. Low-vs-high order components of Lorentz force around a magnetic background
  13. Ill-posedness

Key Result

Theorem 1.1

Let $\sigma,c\in (0,\infty)$, and consider EM on $\mathbb R^2$ or $\mathbb T^2$. There is a universal constant $c_*>0$, depending only on $c$ and $\sigma$, such that, for any $N>0$, there is a smooth initial datum $(u_{0,N}, E_{0,N},B_{0,N})$ satisfying with and generating a unique local-in-time smooth solution $(u_N,E_N,B_N)$ of EM such that for some positive time $t \lesssim 1/N$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 2.1: Product-laws
  • Lemma 2.2: EM20
  • Lemma 2.3: V98
  • Lemma 2.4: bcd11
  • Lemma 2.5
  • proof
  • Lemma 3.1: Energy
  • proof
  • Theorem 3.2: Local smooth solutions
  • ...and 10 more