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Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on 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 4πm. 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 m4. 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:
35Q55NLS-like (nonlinear Schrödinger) equations
35Q60PDEs in connection with optics and electromagnetic theory
82D40Magnetic materials (statistical mechanics)
35B45A priori estimates for solutions of PDE
35B35Stability of solutions of PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
80A20Heat and mass transfer, heat flow
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