Stability for 2D Incompressible MHD Equations With Vertical Magnetic Dissipation and Vertical Velocity Damping Term

This article primarily investigates the well-posedness and stability of the 2D MHD equations with vertical magnetic dissipation under Diophantine condition. The absence of horizontal magnetic dissipation brings significant challenges in estimating nonlinear terms. By employing frequency decomposition of the solutions and using the damping effect of the velocity field component <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , we succeed in driving some precise priori estimates of the solutions. Additionally, we investigate the existence and stability of solutions under Diophantine condition for other dissipation cases. Specifically, our results extend and generalize the corresponding results established in Zhai and Zheng and Zhu.