  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 $\Om$ or a $n$-dimensional
closed Riemannian manifold $M$ into a 2-dimensional unit sphere $\U^{2}$. Our
conclusions extend a series of related results obtained in the previous
literature.
