Liouville type theorems for the 3D stationary MHD and Hall-MHD equations with non-zero constant vectors at infinity

TL;DR

This paper establishes Liouville type theorems for 3D steady-state MHD and Hall-MHD equations, showing under certain asymptotic conditions that the velocity and magnetic fields are constant vectors at infinity.

Contribution

It provides new Liouville theorems for Hall-MHD systems with non-zero boundary conditions at infinity, including cases with degenerate viscosity or resistivity, using advanced $L^p$ estimates and stability analysis.

Findings

$u$ and $B$ are constant vectors when $B$ tends to a non-zero constant at infinity.

$u$ and $B$ are constant when $u$ tends to a constant at infinity and $B$ tends to zero.

Results are obtained under minimal assumptions, extending previous Liouville theorems to more general conditions.

Abstract

In this paper, we investigate Liouville type theorems for the three-dimensional steady-state MHD or Hall-MHD system under some asymptotic assumptions at infinity. Firstly, for the Hall-MHD system we obtain that $u$ and $B$ are constant vectors for any fluid viscosity, magnetic resistivity or Hall-coefficient when the magnetic field $B$ tends to a non-zero constant vector at infinity while the velocity field $u$ tends to $0$. Secondly, it also follows that $u$ and $B$ are constant for the Hall-MHD system when the velocity field tends to a constant vector at infinity while the magnetic field tends to $0$ without any assumptions on viscosity, magnetic resistivity or Hall-coefficient. One main difficulty lies in the Hall term, and we obtain the $L^p$ estimates of a generalized Oseen system with some supercritical terms via Lizorkin's theory and prove that the operator is stable by exploring Kato's stability theorem. Moreover, some similar results for the degenerate fluid viscosity or magnetic resistivity for the MHD system are also obtained, which is independent of interest.