论文标题
具有自然边界条件的Landau-Lifshitz-Gilbert方程的均质化
Homogenization of the Landau-Lifshitz-Gilbert equation with natural boundary condition
论文作者
论文摘要
具有周期性材料系数和自然边界条件的完整Landau-Lifshitz-Gilbert方程式用于对复合铁磁体中的磁化动力学进行建模。在这项工作中，我们通过宽松的等效定理类型的论点建立了均质解和原始解决方案之间的收敛性。有一些技术困难，包括：1）它被证明是均质化的经典选择，不能在$ h^1 $ norm中提供收敛结果； 2）由于自然边界条件，诱导边界层； 3）杂散场的存在引起了多尺度的潜在问题。为了保持收敛速率在边界附近，我们通过高阶修改引入了Neumann校正器。推导了对函数和边界层的奇异积分的估计值，以进行杂散场的一致性分析。此外，受磁化的长度保守性的启发，我们选择了特定几何空间中的适当校正器。这些以及原始解决方案上的均匀$ w^{1,6} $估算的估计值，以$ h^1 $ sense提供收敛速率。
The full Landau-Lifshitz-Gilbert equation with periodic material coefficients and natural boundary condition is employed to model the magnetization dynamics in composite ferromagnets. In this work, we establish the convergence between the homogenized solution and the original solution via a Lax equivalence theorem kind of argument. There are a few technical difficulties, including: 1) it is proven the classic choice of corrector to homogenization cannot provide the convergence result in the $H^1$ norm; 2) a boundary layer is induced due to the natural boundary condition; 3) the presence of stray field give rise to a multiscale potential problem. To keep the convergence rates near the boundary, we introduce the Neumann corrector with a high-order modification. Estimates on singular integral for disturbed functions and boundary layer are deduced, to conduct consistency analysis of stray field. Furthermore, inspired by length conservation of magnetization, we choose proper correctors in specific geometric space. These, together with a uniform $W^{1,6}$ estimate on original solution, provide the convergence rates in the $H^1$ sense.
