# Global well-posedness of magnetohydrodynamic equations

## TL;DR

This paper proves the global existence of solutions and the existence of a uniform attractor for the magnetohydrodynamic equations, advancing understanding of their long-term behavior under specific boundary conditions.

## Contribution

It establishes the global well-posedness and attractor existence for MHD equations with particular boundary conditions, which was previously unresolved.

## Key findings

- Global existence of weak solutions
- Existence of strong solutions
- Existence of a uniform attractor

## Abstract

We study the global well-posedness of magnetohydrodynamic (MHD) equations. The hydrodynamic system consists of the Navier-Stokes equations for the fluid velocity coupled with a reduced from of the Maxwell equations for the magnetic field. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the magnetic field is subject to a time-dependent Dirichlet boundary condition. We first establish the global existence of weak and strong solutions to (1.1)-(1.4). Then we derive the existence of a uniform attractor for (1.1)-(1.4).