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 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 T or a bounded domain in R n into a unit sphere S g (1) in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of 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.

In this paper, we consider the Landau-Lifshitz-Gilbert systems with spin-polarized transport from a bounded domain in R 3 into S 2 and show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz systems. In particular, we show that the Cauchy problem to Landau-Lifshitz equation without damping but with diffusion process of the spin accumulation admits a global weak solution. The Landau-Lifshtiz system with spin-polarized transport into a compact Lie algebra is also discussed and some similar results are proved. The key ingredients of this article consist of the choices of test functions and approximate equations.

In this paper, we consider the high order geometric flows of a submanifolds M in a complete Riemannian manifold N with dim(N ) = dim(M ) + 1 = n + 1, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on N . Precisely, we show that if m ∈ N is strictly larger than the integer part of n/2 and ϕ(t) is a immersion for all t ∈ [0, T ) and if F m (ϕ 0 ) is bounded by a constant which relies on the injectivity radiusR > 0 and sectional curvatureK π (K π 1) of N , then T must be ∞.