Renormalization for the Laplacian and global well-possness of the Landau–Lifshitz–Gilbert equation in dimensions $n \geq3

The global solution of the $n \geq3$ n ≥ 3 Landau–Lifshitz–Gilbert equation on $\mathbb{S}^{2}$ 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 $H^{2}$ H 2 well-posedness of the Cauchy problem for the Landau–Lifshitz–Gilbert equation is established.