##
**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 \({\mathbf u}(x,y,t)\) 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.

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)

### 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 |

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\textit{S. Gustafson} et al., Duke Math. J. 145, No. 3, 537--583 (2008; Zbl 1170.35091)

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