Global Regularity Criteria for Generalized Magneto-Hydrodynamic Systems in Critical Besov Spaces

This monograph investigates the global well-posedness and regularity breakdown criteria for the three-dimensional generalized magneto-hydrodynamic (GMHD) equations with fractional dissipative operators (-) and (-). We establish that for initial data in the critical homogeneous Besov spaces sp,q(R3), the system admits a unique global solution provided the initial norms are sufficiently small relative to the viscosity and magnetic diffusivity. Furthermore, we derive a refined blow-up criterion of Serrin-type, demonstrating that a solution can be extended beyond a maximal time T* if and only if the velocity field satisfies a specific integrability condition in the Besov framework. Our analysis relies on advanced Littlewood-Paley decomposition techniques, commutator estimates in Besov spaces, and the exploitation of the specific algebraic structure of the nonlinear advection and coupling terms.