# Global classical solutions of 3D viscous MHD system without magnetic diffusion on periodic boxes

## TL;DR

This paper proves the global existence of smooth solutions for the 3D viscous MHD system without magnetic diffusion on periodic domains, using energy estimates and symmetry assumptions.

## Contribution

It establishes the first global existence result for 3D viscous MHD without magnetic diffusion under specific initial conditions.

## Key findings

- Global classical solutions exist under initial symmetry and closeness to equilibrium.
- Energy estimates are effective in controlling the system's behavior.
- Results apply to periodic boundary conditions in 3D.

## Abstract

In this paper, we study the global existence of classical solutions to the three dimensional incompressible viscous magneto-hydrodynamical system without magnetic diffusion on periodic boxes, i.e., with periodic boundary conditions. We work in Eulerian coordinate and employ a time-weighted energy estimate to prove the global existence result, under the assumptions that the initial magnetic field is close enough to an equilibrium state and the initial data have some symmetries.