A general Liouville-type theorem for the 3D steady-state Magnetic-B\'enard system

TL;DR

This paper proves a Liouville-type theorem for the 3D steady Magnetic-Bénard system, showing trivial solutions are unique under certain space conditions, extending to various coupled systems and function spaces.

Contribution

It generalizes Liouville theorems to the Magnetic-Bénard system and related models using local Morrey spaces, covering multiple function space settings.

Findings

Uniqueness of trivial solutions under Morrey space conditions.

Extension of Liouville theorems to coupled systems like Boussinesq and MHD-Boussinesq.

Applicability to Lebesgue, Lorentz, Morrey, and weighted-Lebesgue spaces.

Abstract

We establish a Liouville-type theorem for the elliptic and incompressible Magnetic-B\'enard system defined over the entire three-dimensional space. Specifically, we demonstrate the uniqueness of trivial solutions under the condition that they belong to certain local Morrey spaces. Our results generalize in two key directions: firstly, the Magnetic-B\'enard system encompasses other significant coupled systems for which the Liouville problem has not been previously studied, including the Boussinesq system, the MHD-Boussinesq system, and the B\'enard system. Secondly, by employing local Morrey spaces, our theorem applies to Lebesgue spaces, Lorentz spaces, Morrey spaces, and certain weighted-Lebesgue spaces.