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arXiv · 0904.0461

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

S. Gustafson·K. Nakanishi·T. -P. Tsai
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Abstract

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

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S. Gustafson, K. Nakanishi, T. -P. Tsai. 2009-04-03. Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2. https://doi.org/10.1007/s00220-010-1116-6

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