On some new global existence result of 3D Magnetohydrodynamic equations

TL;DR

This paper proves a new global existence result for 3D incompressible Magnetohydrodynamic equations, showing that small initial differences between magnetic field and velocity lead to solutions that persist indefinitely, highlighting magnetic regularization effects.

Contribution

It establishes a global strong solution without smallness restrictions on initial velocity or magnetic field, revealing magnetic field's regularizing influence.

Findings

Global strong solutions exist under small initial magnetic-velocity difference.

Magnetic field can regularize Navier-Stokes equations through cancellation.

No smallness condition needed on initial velocity or magnetic field.

Abstract

This paper is devoted to the incompressible Magenetohydrodynamic equations in $\R^3$. We prove that if the difference between the magnetic field and the velocity is small initially then it will remain forever, thus results in global strong solution without smallness restriction on the size of initial velocity or magnetic field. In other words, magnetic field can indeed regularize the Navier-Stokes equations, due to cancelation.