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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(x,y,t) in ferromagnetic media:

𝐮 t=𝐮×Δ𝐮,

where Δ is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge m>0, are looked for as 𝐮=e imθ 𝐯(r), where r,θ are the polar coordinates in the plane. The vortex solution decays at r, 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 m4. This limitation is imposed by the necessity of a sufficiently quick decay of the unperturbed solution at r. The situation for the vortices with 1m3, 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:
35Q55NLS-like (nonlinear Schrödinger) equations
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B35Stability of solutions of PDE