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Asymptotic stability of harmonic maps under the Schrödinger flow. (English) Zbl 1170.35091

The paper aims to report results concerning the presence or absence of the dynamical collapse (blowup in a finite time) of finite-energy two-dimensional vortex solutions to the Landau-Lifshitz equation, which is fundamental equation governing the dynamics of local magnetization $𝐮\left(x,y,t\right)$ in ferromagnetic media:

$\frac{\partial 𝐮}{\partial t}=𝐮×{\Delta }𝐮,$

where ${\Delta }$ is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge $m>0$, are looked for as $𝐮={e}^{im\theta }𝐯\left(r\right)$, where $r,\theta$ are the polar coordinates in the plane. The vortex solution decays at $r\to \infty$, essentially, as ${r}^{-m}$. First, the work produces a proof of theorems stating the local well-posedness and orbital stability of solutions close to the vortices, but only up to the moment of possible blowup (collapse) of the solutions.

The main result of the work is a theorem which states the absence of the collapse in solutions close to the vortices with $m\ge 4$. This limitation is imposed by the necessity of a sufficiently quick decay of the unperturbed solution at $r\to \infty$. The situation for the vortices with $1\le m\le 3$, and for the zero-vorticity states, with $m=0$, remains unknown. The proofs are based on the decomposition of the solution into the unperturbed part and dispersive perturbations, to which the so-called Strichartz estimates, following from the linearized version of the underlying equation, are applied.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 35B30 Dependence of solutions of PDE on initial and boundary data, parameters 35B35 Stability of solutions of PDE