Communications in Mathematical Physics

, Volume 300, Issue 1, pp 205–242

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


  • Stephen Gustafson
    • Department of MathematicsUniversity of British Columbia
    • Department of MathematicsKyoto University
  • Tai-Peng Tsai
    • Department of MathematicsUniversity of British Columbia

DOI: 10.1007/s00220-010-1116-6

Cite this article as:
Gustafson, S., Nakanishi, K. & Tsai, T. Commun. Math. Phys. (2010) 300: 205. doi:10.1007/s00220-010-1116-6


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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© Springer-Verlag 2010