Ill-posedness of the incompressible magneto-hydrodynamics system in $\dot{B}_{\infty,\infty}^{-1}$

Abstract

We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed in the largest scaling invariant space $\dot{B}_{\infty,\infty}^{-1}$. The construction method of initial data used in this paper is different from the one in a previous work [DQS] of the author. Specifically, we construct initial data which has finite energy and is small enough in $H^s\cap \dot{B}_{\infty,\infty}^{-1}$, with $s<1/2$, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in the metric of $\dot{B}_{\infty,\infty}^{-1}$.