A General Liouville-Type Theorem for the 3D Steady-State Magnetic-Bénard System

Abstract

We establish a Liouville-type theorem for the elliptic, incompressible Magnetic–Bénard system posed on the whole three-dimensional space. In particular, we prove that the only solutions belonging to certain local Morrey spaces are trivial. Our results extend existing theory in two important directions. First, the Magnetic–Bénard system provides a unified framework that includes several fundamental coupled systems for which Liouville-type results have not previously been investigated, such as the Boussinesq system, the MHD–Boussinesq system, and the Bénard system. Second, by working within the setting of local Morrey spaces, our theorem covers a broad class of function spaces, including Lebesgue spaces, Lorentz spaces, Morrey spaces, and certain weighted Lebesgue spaces.