Communications in Mathematical Physics

, Volume 300, Issue 1, pp 205-242

First online:

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

  • Stephen GustafsonAffiliated withDepartment of Mathematics, University of British Columbia
  • , Kenji NakanishiAffiliated withDepartment of Mathematics, Kyoto University Email author 
  • , Tai-Peng TsaiAffiliated withDepartment of Mathematics, University of British Columbia

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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”.