Gustafson, Stephen; Kang, Kyungkeun; Tsai, Tai-Peng Asymptotic stability of harmonic maps under the Schrödinger flow. (English) Zbl 1170.35091 Duke Math. J. 145, No. 3, 537-583 (2008). 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 \({\mathbf u}(x,y,t)\) in ferromagnetic media: \[ \frac{\partial{\mathbf u}}{\partial t}= {\mathbf u}\times\Delta {\mathbf u}, \] where \(\Delta\) is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge \(m>0\), are looked for as \({\mathbf u}=e^{im\theta}{\mathbf v}(r)\), 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.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. Reviewer: Boris A. Malomed (Tel Aviv) Cited in 1 ReviewCited in 29 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B35 Stability in context of PDEs Keywords:Landau-Lifshitz equation; vortex; collapse; Strichartz estimates PDFBibTeX XMLCite \textit{S. Gustafson} et al., Duke Math. J. 145, No. 3, 537--583 (2008; Zbl 1170.35091) Full Text: DOI arXiv References: [1] I. Bejenaru, On Schrödinger maps , Amer. J. Math. 130 (2008), 1033–1065. · Zbl 1159.35065 [2] N. Burq, F. Planchon, J. G. Stalker, and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential , J. Funct. Anal. 203 (2003), 519–549. · Zbl 1030.35024 [3] -, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay , Indiana Univ. Math. J. 53 (2004), 1665–1680. · Zbl 1084.35014 [4] K.-C. Chang, W. Y. Ding, and R. 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