论文标题
Landau-Lifshitz方程的最新结果
Recent results for the Landau-Lifshitz equation
论文作者
论文摘要
我们对有关Landau-Lifshitz方程的一些最新结果进行了调查，Landau-Lifshitz方程是具有强大几何含量的基本非线性PDE，描述了铁磁材料中磁化的动力学。我们回顾了各向异性Landau-Lifshitz方程的库奇问题，而无需耗散，以使解决方案以及在维度一的能量空间中。我们还研究了正弦方程和立方schrödinger方程给出的landau-lifshitz方程的两个近似值，分别在某些奇异的易于平面和易于轴各向异性的界限下产生。 关于局部解决方案，我们回顾了尺寸唯一的孤子总和的轨道和渐近稳定性问题，从而利用了流体动力学框架中孤子的变化性质。 最后，我们调查了有关吉尔伯特（Gilbert）术语的各向同性landau-lifshitz方程的自相似解决方案（扩展器和收缩器）的存在，独特性和稳定性的结果。由于扩展器与奇异的初始条件相关联，因此我们还审查了它们在与BMO空间相关的空间中的良好性。
We give a survey on some recent results concerning the Landau-Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We revisit the Cauchy problem for the anisotropic Landau-Lifshitz equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. We also examine two approximations of the Landau-Lifshitz equation given by of the Sine-Gordon equation and cubic Schrödinger equations, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, we review the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical framework. Finally, we survey results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic Landau-Lifshitz equation with Gilbert term. Since expanders are associated with a singular initial condition with a jump discontinuity, we also review their well-posedness in spaces linked to the BMO space.
