Recent results for the Landau–Lifshitz equation

We give a survey on some recent results concerning the Landau–Lifshitz equation, fundamental nonlinear PDE with strong geometric content, describing dynamics of magnetization in ferromagnetic materials. revisit Cauchy problem for anisotropic LL without dissipation, smooth solutions, and also energy space dimension one. examine two approximations equation given by Sine–Gordon cubic Schrödinger equations, arising certain singular limits easy-plane easy-axis anisotropy, respectively. Concerning localized we review orbital asymptotic stability problems sum solitons one, exploiting variational nature hydrodynamical frameworkFinally, existence, uniqueness self-similar solutions (expanders shrinkers) isotropic Gilbert term. Since expanders are associated initial condition jump discontinuity, their well-posedness spaces linked to BMO space.