| Landau-Lifshitz equations of ferromagnetism, which are based on several competing energy contributions, are important mathematical models for the evolution of magnetization field m of a ferromagnetic material. Many problems, such as existence, stability, regularity, asymptotic behavior, thin-film limit and numerical computation, have been well studied for the Landau-Lifshitz equations that include the so-called exchange energy. However, these problems turn out to be quite challenging for equations without the exchange energy. The main reason is that when the exchange energy is included, one automatically has the magnetization vector m ∈ L ∞((0,∞); H1(Ω)) from energy estimates, which gives some compactness and stability that are needed for using the standard methods; however, in the cases without the exchange energy, one only has m ∈ L∞ ((0, ∞); L∞(Ω)), which is too rough to get the needed compactness and stability. In this thesis, we investigate some problems for models of reduced Landau-Lifshitz equations with no-exchange energy.;In Chapter 1, we introduce the Landau-Lifshitz theory of ferromagnetism and summarize the main results of the thesis. The readers can check out the main results quickly in this chapter and then go to the corresponding chapters for details of proof, more discussions and further references.;In Chapter 2, we study the quasi-stationary limit of a simple Landau-Lifshitz-Maxwell system with the permittivity parameter ε approaching zero and, using this quasi-stationary limit, establish the existence of global weak solutions to the reduced Landau-Lifshitz equations with initial value m0∈ L∞(Ω).;In Chapter 3, we establish a local L2-stability theorem for the global weak solutions in finite time. The key in the proof of stability theorem is that we split the nonlocal term H m into two parts: one is bounded in L ∞(Ω) and the other bounded in L 2(Ω). Using this stability theorem, we also provide another proof for the existence of global weak solutions for a full expression of the no-exchange energy with applied field a(x) ∈ L∞(Ω).;In Chapter 4, we prove a higher time regularity for the regular solutions, using mainly induction method, together with several interpolation results. In this chapter, we also study the weak ω-limit sets for the so-called soft-case and study the asymptotic behaviors for the special case when Ω is ellipsoid and initial values m0 are constant.;In Chapter 5, we investigate a different model called the fractional Landau-Lifshitz equations and establish the existence of global weak solutions with initial value m0 in H α(Ω), where 0 < α < 1 . In this new model, in contrast to the case when only the nonlocal term H m is included, we have some compactness in H α(Ω), which enables us to apply the Galerkin method to establish the existence of global weak solution. |
