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Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on \(\mathbb R^2\). (English) Zbl 1205.35294

The paper considers the Landau-Lifshitz (LL) equation which governs the evolution of the local magnetization in the 2D space. The equation also includes a dissipative term. It is known that various configurations of the 3D magnetization vector with a fixed length in the 2D space, i.e., the map of the 2D space into the 2D sphere, may be classified by an integer index \(m\) (“degree”). The energy of each subclass of the solutions, pertaining to a particular value of \(n\), is limited from below by the minimum value \(4\pi m\). In previous works, it was demonstrated that the solutions of the 2D LL equation with energies taken close enough to the \(m\)-dependent minimum are stable for \(m\geq 4\). In the present work, the same statement is proved for \(m=3\). The analysis is also extended for more particular (not most generic) types of the solutions pertaining to \(m=2\), which is much harder for the consideration, as in this case, it is difficult to decompose the general solution into an essential part and dispersing perturbations. In the present work, this is done by means of a specially devised nonorthogonal decomposition. It is concluded that the solutions for \(m=2\) may demonstrate different types of the dynamical behavior: asymptotic stability, collapse (the formation of a singularity), and persistent oscillations.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
82D40 Statistical mechanics of magnetic materials
35B45 A priori estimates in context of PDEs
35B35 Stability in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
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