Global Existence and Decay Estimates of the Three Dimensional Magneto-Hydrodynamic Equations with Partial Dissipation

Abstract

To uncover the mechanism by which magnetic fields can stabilize electrically conducting turbulent fluids, we investigate the stability of a special three-dimensional anisotropic magnetohydrodynamic (MHD) system. This system features dissipation only in the direction of \(x_{1}\) and magnetic damping near a background magnetic field. Due to the absence of dissipation in the \(x_{2}\) and \(x_{3}\) directions, establishing the stability and long-time behavior of this MHD system is highly nontrivial. Through a subtle energy estimate and careful analysis of the nonlinearities, we rigorously justify the stability of this MHD system near a background magnetic field and derive explicit decay rates. The primary challenge lies in proving the uniform integrability of \(\|\nabla _{h}u\|_{L^{\infty }}\) in the time variable.