On stability and asymptotic behaviours for a degenerate Landau-Lifshitz equation
نویسنده
چکیده
In this paper, we study the problem concerning stability and asymptotic behaviours of solutions for a degenerate Landau-Lifshitz equation in micromagnetics involving only the nonlocal magnetostatic energy. Due to the lack of derivative estimates, we do not have the compactness needed for strong convergence and the natural convergence is weak* convergence. By formulating the problem in a new framework of differential inclusions, we show that the Cauchy problems for such an equation are not stable under the weak* convergence of initial data. For the asymptotic behaviours of weak solutions, we establish an estimate on the weak* ω-limit sets that is valid for all initial data satisfying the saturation condition.
منابع مشابه
On Landau-lifshitz Equations of No-exchange Energy Models in Ferromagnetics
In this paper, we study Landau-Lifshitz equations of ferromagnetism with a total energy that does not include a so-called exchange energy. Many problems, including existence, stability, regularity and asymptotic behaviors, have been extensively studied for such equations of models with the exchange energy. Problems turn out quite different and challenging for Landau-Lifshitz equations of no-exc...
متن کاملLie Group Classifications and Stability of Exact Solutions for Multidimensional Landau-Lifshitz Equations
In this paper, based on classical Lie group method, we study multi-dimensional Landau-Lifshitz equation, and get its infinitesimal generator, symmetry group and new solutions. In particular, we build the connection between new exact solutions and old exact solutions. At the same time, we also prove that the initial boundary value condition of the three-dimensional Landau-Lifshitz equation admit...
متن کاملQuasi-stationary Limit and a Landau-lifshitz Equation in Ferromagnetism
In this paper we study a Landau-Lifshitz equation in ferromagnetism without exchange energy as a quasi-stationary limit of a coupled Maxwell system as dielectric permittivity approaches zero. We present a different formulation and proof of this quasi-stationary limit result. We also prove a new uniqueness and stability result for the limit Landau-Lifshitz equation.
متن کاملAsymptotic Stability , Concentration , and Oscillation in Harmonic Map Heat - Flow , Landau - Lifshitz , and Schrödinger Maps on R 2
We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from R2 to S2. If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work (Gustafson et al., Duke Math J 145(3), 537–583, 2008) down to degree m = 3. D...
متن کاملMinimal Energy for the Traveling Waves of the Landau-Lifshitz Equation
We consider nontrivial finite energy traveling waves for the Landau–Lifshitz equation with easyplane anisotropy. Our main result is the existence of a minimal energy for these traveling waves, in dimensions two, three and four. The proof relies on a priori estimates related with the theory of harmonic maps and the connection of the Landau–Lifshitz equation with the kernels appearing in the Gros...
متن کامل