Liouville Type Theorem for the Fractional MHD and Hall-MHD equations in $\mathbb{R}^{3}

TL;DR

This paper establishes Liouville type theorems for stationary fractional MHD and Hall-MHD equations in three-dimensional space, utilizing extension techniques to handle non-local operators, and also provides results for the Navier-Stokes equations.

Contribution

It introduces new Liouville theorems for fractional MHD and Hall-MHD equations using the Caffarelli-Silvestre extension method, extending previous results to non-local fractional operators.

Findings

Liouville theorems for fractional MHD and Hall-MHD equations.

Extension method effectively handles non-local fractional Laplacian.

Additional results obtained for the Navier-Stokes equations.

Abstract

In this paper, we are mainly concerned with the Liouville type problem for the stationary fractional magnetohydrodynamics(MHD) and stationary fractional Hall-MHD equations. In addition, we present the results of the Navier-Stokes equation as a byproduct. The key point is to use the Caffarelli-Sivestre extension to overcome the difficulty caused by the non-local operator $(-\triangle)^{s}$ and combined with Yuan and Xiao's method (J. Math. Anal. Appl. 491 (2020) 124343).