Global Small Solutions to an MHD-Type System: The Three-Dimensional Case

In this paper, we consider the global well-posedness of a three-dimensional incompressible MHD type system with smooth initial data that is close to some nontrivial steady state. It is a coupled system between the Navier-Stokes equations and a free transport equation with a universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of φ{symbol} in the coupled equations and only the horizontal derivatives of φ{symbol} is dissipative with respect to time. To achieve this, we first employ anisotropic Littlewood-Paley analysis to establish the key L¹(ℝ⁺; Lip(ℝ³)) estimate to the third component of the velocity field. Then we prove the global well-posedness to this system by the energy method, which depends crucially on the divergence-free condition of the velocity field.