论文标题
薄膜微磁化学中边界涡流的有效模型
An effective model for boundary vortices in thin-film micromagnetics
论文作者
论文摘要
铁磁材料由非本地，非凸和多尺度的变异原理约束。主要对象由单位长度的三维矢量场给出，磁化化对应于微磁能的稳定状态。我们的目的是分析一种薄膜状态，该状态捕获磁化强度及其相互作用能产生的边界涡流的渐近行为。这项研究基于“全球雅各布”的概念，该概念检测到拓扑缺陷，即先前可以位于膜的内部和边界。一个主要的困难在于估计微磁能的非局部部分，以隔离与拓扑缺陷相对应的确切术语。我们通过二阶通过$γ$ - convergence膨胀来证明边界涡流周围的能量浓度。第二阶项是重新归一化的能量，代表边界涡流之间的相互作用并控制其最佳位置。我们计算了重新归一化能量的表达，我们证明了具有两个具有多重性$ 1 $的边界涡旋的最小化器的存在。还显示了磁化和相应的全局雅各布的紧凑性结果。
Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of "global Jacobian" detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a $Γ$-convergence expansion at the second order. The second order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity $1$. Compactness results are also shown for the magnetization and the corresponding global Jacobian.
