The global solution of the $n \geq3$ Landau–Lifshitz–Gilbert equation on $\mathbb{S}^{2}$ is studied under the cylindrical symmetric coordinates. An equivalent complex-valued equation in cylindrical symmetric coordinates is obtained by the Hasimoto transformation. A renormalization for the Laplacian is used to transform this equivalent system to a Ginzberg–Landau type system in which the Strichartz estimate can be applied. The global $H^{2}$ well-posedness of the Cauchy problem for the Landau–Lifshitz–Gilbert equation is established.

中文翻译：

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尺寸\（n \ geq3 \）中的Landau-Lifshitz-Gilbert方程的Laplacian和整体井然性的重新正规化

在圆柱对称坐标下研究了$ \ mathbb {S} ^ {2} $上的$ n \ geq3 $ Landau–Lifshitz–Gilbert方程的整体解。通过Hasimoto变换获得圆柱对称坐标中的等效复数值方程。拉普拉斯算子的重新规范化用于将该等效系统转换为可以应用Strichartz估计的Ginzberg–Landau类型系统。建立了Landau–Lifshitz–Gilbert方程的Cauchy问题的整体$ H ^ {2} $适定性。