Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}_{\infty}^{-1,\infty}$

TL;DR

This paper shows that solutions to the 3D incompressible MHD system can exhibit rapid norm inflation in certain Besov spaces, with magnetic or velocity fields inflating independently in short time frames.

Contribution

It extends the Bourgain-Pavlović construction from Navier-Stokes to the MHD system, demonstrating norm inflation phenomena in Besov space $ abla_{ ext{B}_{ ext{infty}}}^{-1, ext{infty}}$.

Findings

Magnetic field can inflate in norm rapidly while velocity remains small.

Velocity can inflate independently of magnetic field.

Provides an expository development of the Bourgain-Pavlović construction for MHD.

Abstract

We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to present a very expository development of the inginious construction of Bourgain and Pavlovi\'{c} for Navier-Stokes equations.