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Zbl 1205.35294
Gustafson, Stephen; Nakanishi, Kenji; Tsai, Tai-Peng
Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on $\Bbb R^2$. (English)
[J] Commun. Math. Phys. 300, No. 1, 205-242 (2010). ISSN 0010-3616; ISSN 1432-0916/e

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.
[Boris A. Malomed (Tel Aviv)]

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
35Q60 PDE of electromagnetic theory and optics
82D40 Magnetic materials
35B45 A priori estimates
35B35 Stability of solutions of PDE
35B05 General behavior of solutions of PDE
80A20 Heat and mass transfer

Keywords: Strihartz estimates; non-orthogonal decomposition; asymptotic stability; collapse; Landau-Lifshitz equation

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Zentralblatt MATH Berlin [Germany]