Stability and sharp decay for 3D incompressible MHD system with fractional horizontal dissipation and magnetic diffusion

Abstract

This paper aims as the stability and large-time behavior of 3D incompressible magnetohydrodynamic (MHD) equations with fractional horizontal dissipation and magnetic diffusion. By using the energy methods, we obtain that if the initial data are small enough in $$H^3(\mathbb {R}^3)$$ , then this system possesses a global solution, and whose horizontal derivatives decay at least at the rate of $$(1+t)^{-\frac{1}{2}}$$ . Moreover, if we control the initial data further small in $$H^3(\mathbb {R}^3)\cap H_h^{-1}(\mathbb {R}^3)$$ , the sharp decay of this solution and its first-order derivatives is established.