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

TL;DR

This paper demonstrates that the well-posedness of the 2D incompressible Euler--Maxwell system in the Yudovich class critically depends on the Normal Structure condition, showing ill-posedness when this condition is not met.

Contribution

It proves that the Normal Structure condition is necessary for well-posedness of the Euler--Maxwell system in the Yudovich class, establishing ill-posedness without this condition.

Findings

Ill-posedness occurs without the Normal Structure condition.

The proof applies to both the plane and the torus.

The analysis holds for any finite speed of light c.

Abstract

It was shown recently by Ars\'enio 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.