# Global classical solutions of 2D MHD system with only magnetic diffusion on periodic domain

## TL;DR

This paper proves the global existence of classical solutions for a 2D incompressible MHD system with magnetic diffusion on a periodic domain, using energy estimates under symmetric initial conditions.

## Contribution

It establishes the global well-posedness of the 2D MHD system with magnetic diffusion under symmetry and proximity assumptions, which was previously unresolved.

## Key findings

- Global classical solutions exist under certain initial conditions.
- The method relies on time-weighted energy estimates.
- Symmetry and closeness to equilibrium are key assumptions.

## Abstract

This paper studies the global existence of classical solutions to the two-dimensional incompressible magneto-hydrodynamical (MHD) system with only magnetic diffusion on the periodic domain. The approach is based on a time-weighted energy estimate, under the assumptions that the initial magnetic field is close enough to an equilibrium state and the initial data have reflection symmetry.