Global Existence for Two Dimensional Incompressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity

TL;DR

This paper proves the global existence of classical solutions for 2D incompressible MHD flows with zero magnetic diffusivity, using a perturbation approach and novel techniques to handle the degenerated system.

Contribution

It introduces a new method involving the deformation gradient to decouple flow and magnetic field coupling in zero diffusivity MHD equations.

Findings

Established global-in-time classical solutions in critical spaces.

Achieved $L^1$ dissipation of velocity.

Handled degenerated parabolic-hyperbolic system effectively.

Abstract

The existence of global-in-time classical solutions to the Cauchy problem of incompressible Magnetohydrodynamic flows with zero magnetic diffusivity is considered in two dimensions. The linearization of equations is a degenerated parabolic-hyperbolic system. The solution is constructed as a small perturbation of a constant background in critical spaces. The deformation gradient has been introduced to decouple the subtle coupling between the flow and the magnetic field. The $L^1$ dissipation of the velocity is obtained.