论文标题
在纳米线中使用dzyaloshinskii-Moriya相互作用的landau-lifshitz-gilbert方程的进攻域壁的渐近稳定性
Asymptotic stability of precessing domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction
论文作者
论文摘要
我们考虑了铁磁纳米线，我们专注于渐近方案，其中考虑了dzyaloshinskii-moriya的相互作用。首先，我们通过$γ$ -Convergence证明了缩小尺寸结果，该结果确定了为映射$ M的限制功能$ e $：\ Mathbb {r} \ to \ Mathbb {s}^2 $在方向上$ e_1 $ e_1 $ e_1 $。能量功能$ e $在$ e_1 $的翻译下不变，围绕轴$ e_1 $的旋转。当实施$ -e_1 $和$ e_1 $之间的过渡时，我们将有限能源$ e $的关键点完全分类；这些过渡层称为（静态）域壁。在施加的磁场$ h（t）e_1 $的效果下，与$ e $相关的landau-lifshitz-gilbert方程的域墙的演变，具体取决于时间变量$ t $，从而增加了所谓的预制域壁。我们的主要结果证明了针对小$ h $ in $ l^\ infty（[[0， +\ infty））$和小$ h^1（\ Mathbb {r}）$扰动的静态域墙壁的渐近稳定性。
We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First we prove a dimension reduction result via $Γ$-convergence that determines a limit functional $E$ defined for maps $m:\mathbb{R}\to \mathbb{S}^2$ in the direction $e_1$ of the nanowire. The energy functional $E$ is invariant under translations in $e_1$ and rotations about the axis $e_1$. We fully classify the critical points of finite energy $E$ when a transition between $-e_1$ and $e_1$ is imposed; these transition layers are called (static) domain walls. The evolution of a domain wall by the Landau-Lifshitz-Gilbert equation associated to $E$ under the effect of an applied magnetic field $h(t)e_1$ depending on the time variable $t$ gives rise to the so-called precessing domain wall. Our main result proves the asymptotic stability of precessing domain walls for small $h$ in $L^\infty([0, +\infty))$ and small $H^1(\mathbb{R})$ perturbations of the static domain wall, up to a gauge which is intrinsic to invariances of the functional $E$.
