铁磁链方程（组）中的某些数学问题研究
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摘要
本论文研究铁磁链方程(组)及其相关模型的一些数学问题。铁磁链方程是于1935年由物理学家Landau和Lifshitz在研究铁磁体磁导率的色散理论时提出来的。这是一类很重要的磁化运动方程,经常出现在凝聚态物理的研究中。随着理论研究的不断深入,物理学家近年来提出了一些更加精细的模型。如提出了描述亚铁盐材料的磁化运动方程;在铁磁体材料的研究中考虑了自旋极化输运效应;以及将由声子、输运电子、核自旋等引起的随机效应考虑进Landau-Lifshitz方程以解释磁矩方向的涨落,从而得到相应的具有乘积噪声的非线性随机微分方程。在这篇论文中,我们将从偏微分方程理论的角度去严格证明这些模型在一定意义下的解的整体存在唯一性,并考虑了解的某些渐近性质。特别地我们首次得到了随机Landau-Lifshitz方程一维情形光滑解的整体存在唯一性以及二维和三维情形时小初值光滑解的整体存在唯一性。这些结论是目前为止我们所知的关于这些方程组的最佳结论,对进一步深入研究铁磁体理论是重要的。
第一章是绪论。着重介绍本文研究的物理背景、已有结果、最新进展以及本文的主要结果。
第二章研究一类反铁磁问题的解的存在唯一性。利用惩罚方法,通过构造相应的惩罚问题,得到了该模型弱解的整体存在性;同时利用不动点理论以及先验估计得到了光滑解的整体存在唯一性。更为重要的是我们建立了这类方程和波映照之间的联系。
第三章着重讨论带自旋极化的Landau-Lifshitz方程光滑解的整体存在性。利用精细的先验估计得到了光滑解的整体存在唯一性等结论。
第四章证明不具有Gilbert项时,Landau-Lifshitz方程vortex解的不存在性。
第五章重点讨论具有乘积噪声的随机Landau-Lifshitz方程光滑解的整体存在唯一性。指出了为了得到“热动力学相容性”,该方程的随机积分必须在Stratonovich意义下理解,并在此基础上利用差分方法以及It(?)公式得到了光滑解的整体存在唯一性。最后还讨论了解的爆破现象,并提出了相应的修正模型。
This dissertation concerns the mathematical aspects for equations of ferromagnetic chains and the related models. Ferromagnetic chain equation was first proposed in 1935 by physicists L.D. Landau and E.M. Lifshitz when studying the dispersive theory for magnetic conductivity in magnetic materials. It is an important dynamical equation of magnetization and frequently appears in condensation physics. Very recently some refined models were proposed by physicists to depict the magnetic dynamics in ferrimagnetic materials and, spin polarization transport effects were taken into account in ferromagnetic materials as well. Random effects, caused by such as photons, conducting electrons and nuclear spins, were also introduced into the Landau-Lifshitz equation to account for fluctuations of the magnetic moment orientation, which leads to the highly nonlinear multiplicative stochastic Landau-Lifshitz equation. We prove rigorously in mathematics the existence and uniqueness of solutions in some sense for these models as well as their asymptotic behaviors. In particular, we obtain the existence and uniqueness for the first time for the stochastic Landau-Lifshitz equation in one dimension and the existence and uniqueness of small solutions for spatial dimension two and three. As far as we know, this is the first mathematical result for SLLE and is important for further studies.
In Chapter 1, we briefly introduce the physical background, historic results and the main results in our dissertation.
In Chapter 2, we consider the existence of weak solutions for the dynamical equation in ferrimagnetic materials by penalty methods and Galerkin's approximation. By fixed point theorem and some a priori estimates the existence and uniqueness for smooth solutions are obtained and more importantly we establish the relationship between these equations and the classical wave maps.
In Chapter 3, we focus on the spin-polarized transport equation in dimension 2, for which we get the existence and uniqueness of smooth solutions by inverse function theorem and some a priori estimates.
In Chapter 4, nonexistence of vortex solutions for Landau-Lifshitz equation without Gilbert damping term is obtained.
In Chapter 5, we are concerned with the multiplicative stochastic Landau-Lifshitz equation. It is pointed out that the Stratonovich stochastic integral should be utilized to get the proper thermal consistency, based on which the smooth solutions are obtained via the difference method and the It(o|^) formula. Also some blow up phenomena are discussed in this chapter.
第一章是绪论。着重介绍本文研究的物理背景、已有结果、最新进展以及本文的主要结果。
第二章研究一类反铁磁问题的解的存在唯一性。利用惩罚方法,通过构造相应的惩罚问题,得到了该模型弱解的整体存在性;同时利用不动点理论以及先验估计得到了光滑解的整体存在唯一性。更为重要的是我们建立了这类方程和波映照之间的联系。
第三章着重讨论带自旋极化的Landau-Lifshitz方程光滑解的整体存在性。利用精细的先验估计得到了光滑解的整体存在唯一性等结论。
第四章证明不具有Gilbert项时,Landau-Lifshitz方程vortex解的不存在性。
第五章重点讨论具有乘积噪声的随机Landau-Lifshitz方程光滑解的整体存在唯一性。指出了为了得到“热动力学相容性”,该方程的随机积分必须在Stratonovich意义下理解,并在此基础上利用差分方法以及It(?)公式得到了光滑解的整体存在唯一性。最后还讨论了解的爆破现象,并提出了相应的修正模型。
This dissertation concerns the mathematical aspects for equations of ferromagnetic chains and the related models. Ferromagnetic chain equation was first proposed in 1935 by physicists L.D. Landau and E.M. Lifshitz when studying the dispersive theory for magnetic conductivity in magnetic materials. It is an important dynamical equation of magnetization and frequently appears in condensation physics. Very recently some refined models were proposed by physicists to depict the magnetic dynamics in ferrimagnetic materials and, spin polarization transport effects were taken into account in ferromagnetic materials as well. Random effects, caused by such as photons, conducting electrons and nuclear spins, were also introduced into the Landau-Lifshitz equation to account for fluctuations of the magnetic moment orientation, which leads to the highly nonlinear multiplicative stochastic Landau-Lifshitz equation. We prove rigorously in mathematics the existence and uniqueness of solutions in some sense for these models as well as their asymptotic behaviors. In particular, we obtain the existence and uniqueness for the first time for the stochastic Landau-Lifshitz equation in one dimension and the existence and uniqueness of small solutions for spatial dimension two and three. As far as we know, this is the first mathematical result for SLLE and is important for further studies.
In Chapter 1, we briefly introduce the physical background, historic results and the main results in our dissertation.
In Chapter 2, we consider the existence of weak solutions for the dynamical equation in ferrimagnetic materials by penalty methods and Galerkin's approximation. By fixed point theorem and some a priori estimates the existence and uniqueness for smooth solutions are obtained and more importantly we establish the relationship between these equations and the classical wave maps.
In Chapter 3, we focus on the spin-polarized transport equation in dimension 2, for which we get the existence and uniqueness of smooth solutions by inverse function theorem and some a priori estimates.
In Chapter 4, nonexistence of vortex solutions for Landau-Lifshitz equation without Gilbert damping term is obtained.
In Chapter 5, we are concerned with the multiplicative stochastic Landau-Lifshitz equation. It is pointed out that the Stratonovich stochastic integral should be utilized to get the proper thermal consistency, based on which the smooth solutions are obtained via the difference method and the It(o|^) formula. Also some blow up phenomena are discussed in this chapter.
引文
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