Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional

We follow the idea of Wang \cite{W} to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a $n$-dimensional Euclidean domain $\text{Om}$ or a $n$-dimensional closed Riemannian manifold $M$ into a 2-dimensional unit sphere ${\text{U}}^2$. Our conclusions extend a series of related results obtained in the previous literature.