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

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.