Ill-posedness of the incompressible Euler-Maxwell equations in the Yudovich class

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 ∈ ( 0 , ∞ ) 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.