Asymptotic Stability of Precessing Domain Walls for the Landau–Lifshitz–Gilbert Equation in a Nanowire with Dzyaloshinskii–Moriya Interaction

We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First prove dimension reduction result via $\Gamma$-convergence that determines limit functional $E$ defined for maps $m:\mathbb{R}\to \mathbb{S}^2$ in direction $e_1$ of nanowire. The energy invariant under translations rotations about axis $e_1$. fully classify critical points finite when transition between $-e_1$ imposed; these layers are called (static) domain walls. evolution wall by Landau-Lifshitz-Gilbert equation associated to effect applied magnetic field $h(t)e_1$ depending time variable $t$ gives rise so-called precessing wall. Our main proves stability walls small $h$ $L^\infty([0, +\infty))$ $H^1(\mathbb{R})$ perturbations static wall, up gauge which intrinsic invariances $E$.