具有Dzyaloshinskii-Moriya相互作用的纳米线中Landau-Lifshitz-Gilbert方程进动畴壁的渐近稳定性
Asymptotic stability of precessing domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction
AI总结：
本文通过Γ-收敛和能量泛函分类，证明了铁磁纳米线中Dzyaloshinskii-Moriya相互作用下进动畴壁在磁场扰动下的渐近稳定性。
摘要：
我们考虑一个铁磁纳米线，并关注一个考虑Dzyaloshinskii-Moriya相互作用的渐近区域。首先，我们通过Γ-收敛证明了一个维数约化结果，该结果确定了一个极限泛函E，该泛函定义在沿纳米线方向e1的映射m:R→S^2上。能量泛函E在e1方向的平移和绕e1轴的旋转下不变。当在-e1和e1之间施加转变时，我们完全分类了有限能量E的临界点；这些转变层被称为（静态）畴壁。在施加依赖于时间变量t的外加磁场h(t)e1的影响下，由与E相关的Landau-Lifshitz-Gilbert方程驱动的畴壁演化产生了所谓的进动畴壁。我们的主要结果证明了对于小的h在L^∞([0, +∞))中以及静态畴壁的H^1(R)小扰动，进动畴壁的渐近稳定性，直至一个由泛函E的不变性所固有的规范。
英文摘要：
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 $E$.
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