Asymptotic Behavior of Solutions for the Three-dimensional Generalized Incompressible MHD Equations with Nonlinear Damping Terms

This work is concerned with the three-dimensional generalized incompressible MHD equations with nonlinear damping terms (polynomial damping and exponential damping). We first establish the existence and uniqueness, and then we study the asymptotic behavior of weak solutions for these systems via attractors. The novelty is that the strength of nonlinearities and the degree of dissipations can work together to yield the global existence and uniqueness of the weak solutions of these systems. Since our systems might be not uniqueness, we could not use directly the classical scheme of the dynamical system to find attractors. Therefore, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and its properties.