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We consider the Landau-Lifshitz equations of ferromagnetism (including theharmonic map heat-flow and Schroedinger flow as special cases) for degree mequivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimalenergy solutions converge to a harmonic map as t goes to infinity (asymptoticstability), extending previous work down to degree m = 3. Due to slow spatialdecay of the harmonic map components, a new approach is needed for m=3,involving (among other tools) a "normal form" for the parameter dynamics, andthe 2D radial double-endpoint Strichartz estimate for Schroedinger operatorswith sufficiently repulsive potentials (which may be of some independentinterest). When m=2 this asymptotic stability may fail: in the case ofheat-flow with a further symmetry restriction, we show that more exoticasymptotics are possible, including infinite-time concentration (blow-up), andeven "eternal oscillation".

Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2