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Zbl 1170.35091
Gustafson, Stephen; Kang, Kyungkeun; Tsai, Tai-Peng
Asymptotic stability of harmonic maps under the Schrödinger flow. (English)
[J] Duke Math. J. 145, No. 3, 537-583 (2008). ISSN 0012-7094

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 ${\bold u}(x,y,t)$ in ferromagnetic media: $$ \frac{\partial{\bold u}}{\partial t}= {\bold u}\times\Delta {\bold u}, $$ where $\Delta$ is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge $m>0$, are looked for as ${\bold u}=e^{im\theta}{\bold v}(r)$, 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. \par The main result of the work is a theorem which states the absence of the collapse in solutions close to the vortices with $m\geq 4$. 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 $1\leq m\leq 3$, 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.
[Boris A. Malomed (Tel Aviv)]

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
35B30 Dependence of solutions of PDE on initial and boundary data
35B35 Stability of solutions of PDE

Keywords: Landau-Lifshitz equation; vortex; collapse; Strichartz estimates

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Zentralblatt MATH Berlin [Germany]