Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrodinger Maps on R-2

作者：Gustafson Stephen^*; Nakanishi Kenji; Tsai Tai Peng

来源：Communications in Mathematical Physics, 2010, 300(1): 205-242.

DOI：10.1007/s00220-010-1116-6

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrodinger flow as special cases) for degree m equivariant maps from R-2 to S-2. If m >= 3, we prove that near-minimal energy solutions converge to a harmonic map as t -> a (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 Schrodinger 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".

• 出版日期2010-11