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Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on \({\mathbb R^2}\)

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Abstract

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from \({\mathbb {R}^2}\) to \({\mathbb {S}^2}\) . If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work (Gustafson et al., Duke Math J 145(3), 537–583, 2008) 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 Schrödinger 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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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Stephen Gustafson & Tai-Peng Tsai

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Kenji Nakanishi

Authors
  1. Stephen Gustafson
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  2. Kenji Nakanishi
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  3. Tai-Peng Tsai
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Correspondence to Kenji Nakanishi.

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Communicated by P. Constantin

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Gustafson, S., Nakanishi, K. & Tsai, TP. Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on \({\mathbb R^2}\) . Commun. Math. Phys. 300, 205–242 (2010). https://doi.org/10.1007/s00220-010-1116-6

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  • DOI: https://doi.org/10.1007/s00220-010-1116-6

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