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 in ferromagnetic media:
where is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge , are looked for as , where are the polar coordinates in the plane. The vortex solution decays at , essentially, as . 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 . This limitation is imposed by the necessity of a sufficiently quick decay of the unperturbed solution at . The situation for the vortices with , and for the zero-vorticity states, with , 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.