The three-dimensional Hall magneto-hydrodynamics equations with fractional diffusion and damping: Well-posedness, attractors, and asymptotic determining modes

This paper is dedicated to the global well-posedness and long-time behavior of solutions of the three-dimensional Hall magneto-hydrodynamics equations with fractional diffusion and damping. Our purpose is to understand how the degree of dissipation and damping affect the long-time behavior of the weak solutions. First, we investigate the existence and uniqueness of weak solutions. Then, we explore the asymptotic behavior of weak solutions through attractors. Since our system may not always have a unique solution, we utilize a new framework developed by Cheskidov and Lu called the evolutionary system to obtain various attractors and their properties. Finally, we examine asymptotic determining modes for weak solutions. Our results improve and extend previous ones.