Norm inflation for incompressible magneto-hydrodynamic system ]{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 experience rapid norm inflation in certain Besov spaces, with magnetic or velocity fields inflating independently in short time.

Contribution

It extends the concept of norm inflation to the MHD system, demonstrating independent inflation of magnetic and velocity fields in specific function spaces.

Findings

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

Velocity can inflate in norm independently of magnetic field.

The construction adapts Bourgain and Pavlović's method for Navier-Stokes to 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 equation.