The Landau-Lifshitz equation of the ferromagnetic spin chain and Oseen-Frank flow

In this paper, we consider the Landau-Lifshitz equation of the ferromagnetic spin chain from $\R^2$ to the unit sphere $S^2$ under the general Oseen-Frank energy. We obtain global existence and uniqueness of weak solutions for large energy data; moreover, the number of singular points is finite

Partial regularity to the Landau-Lifshitz equation with spin accumulation

In this paper, we consider a model for the spin-magnetization system that takes into account the diffusion process of the spin accumulation. This model consists of the Landau-Lifshitz equation describing the precession of the magnetization, coupled with a quasi-linear parabolic equation describing the diffusion of the spin accumulation. This paper establishes the global existence and uniqueness of weak solutions for large initial data in $\Bbb R^2$. Moreover, partial regularity is shown. In particular, the solution is regular on $\Bbb R^2\times(0,\infty)$ with the exception of at most finite singular points

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 \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

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