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.

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 $H^s(\mathbb{R}^d)$, $s>1+\frac{d}{2}$.