# The continuous dependence for the Hal-MHD equations with fractional magnetic diffusion

## TL;DR

This paper proves that solutions to the incompressible Hall-MHD equations with fractional magnetic diffusion vary continuously with initial conditions in a specific Sobolev space, ensuring stability of solutions.

## Contribution

It establishes the continuous dependence of solutions on initial data for the Hall-MHD system with fractional magnetic diffusion in certain Sobolev spaces.

## Key findings

- Solutions depend continuously on initial data in $H^s$ for $s > 1 + d/2$
- Provides mathematical validation for stability of solutions
- Extends previous results to fractional magnetic diffusion cases

## Abstract

In this paper we show that the solutions to the incompressible Hall-MHD system with fractional magnetic diffusion depend continuously on the initial data in $H^s(\mathbb{R}^d)$, $s>1+\frac{d}{2}$.