Renormalization for the Laplacian and global well-possness of the Landau–Lifshitz–Gilbert equation in dimensions n ≥ 3 $n \geq3$

Abstract

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

Keywords

• Landau–Lifshitz equation
• Well-posedness
• Hasimoto transformation