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

TL;DR

This paper proves that the 3D incompressible MHD system is ill-posed in the critical space ot;B_{\u221e,\u221e}^{-1}, showing solutions can be discontinuous at initial time despite small initial data.

Contribution

It introduces a new construction of initial data demonstrating ill-posedness of the MHD system in the largest scaling invariant space, differing from previous methods.

Findings

Solutions are discontinuous at t=0 in ot;B_{\u221e,}^{-1} space.

Initial data with finite energy can lead to ill-posedness.

The method differs from previous constructions, highlighting the space's critical nature.

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 $\dot{B}_{\infty,\infty}^{-1}$.