Global Weak Solutions to Landau-Lifshitz Equations into Compact Lie Algebras

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz equations from an $n$-dimensional closed Riemannian manifold $\mathbb{T}$ or a bounded domain in $\mathbb{R}^n$ into a unit sphere $S_\mathfrak{g}(1)$ in $\mathfrak{g}$. In particular, we consider the Hamiltonian system associated with the nonlocal energy--{\it micromagnetic energy} defined on a bounded domain of $\mathbb{R}^3$ and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.