# Global Strong Solution of a 2D coupled Parabolic-Hyperbolic Magnetohydrodynamic System

## TL;DR

This paper proves the existence of global strong solutions for a 2D coupled parabolic-hyperbolic magnetohydrodynamic system, advancing mathematical understanding of MHD models in fluid dynamics.

## Contribution

It establishes the global strong solution existence for a 2D coupled parabolic-hyperbolic MHD system using advanced PDE estimates, which was previously unresolved.

## Key findings

- Existence of global strong solutions in 2D MHD system
- Application of Agmon-Douglis-Nirenberg estimates
- Use of Solonnikov's $L^p$-$L^q$ estimates for evolution Stokes equations

## Abstract

The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic (MHD) model in two dimensional space. Based on Agmon, Douglis and Nirenberg's estimates for the stationary Stokes equation and the Solonnikov's theorem of $L^p$-$L^q$-estimates for the evolution Stokes equation, it is shown that the mixed-type MHD equations exist a global strong solution.