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

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 Pavlović for Navier-Stokes equations.