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

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 R 2 to S 2 . If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work [11] 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".

Blow up and near soliton dynamics for the L 2 critical gKdV equation

“…In particular, other type of blow up speeds can be produced by similar arguments by changing the tail of the initial data. A similar phenomenon was observed for global in time growing up solutions to the parabolic energy critical harmonic heat flow by Gustafson, Nakanishi and Tsai [7]. In [7], an explicit formula on the growth of the solution at infinity is given directly in terms of the initial data.…”

“…A similar phenomenon was observed for global in time growing up solutions to the parabolic energy critical harmonic heat flow by Gustafson, Nakanishi and Tsai [7]. In [7], an explicit formula on the growth of the solution at infinity is given directly in terms of the initial data. Continuums of blow up rates were also observed in pioneering works by Krieger, Schlag and Tataru [15], [16] for energy critical wave problems, see also Donninger and Krieger [3].…”

“…Séminaire Laurent-Schwartz -EDP et applications Institut des hautes études scientifiques, 2011-2012 Exposé n o XXXVII, [1][2][3][4][5][6][7][8][9][10][11][12][13][14] XXXVII-1…”

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“…Asymptotic stability of harmonic maps of topological degree |m| ≥ 4 under the Schrödinger flow is established in [Gustafson et al 2008]. The result is extended to maps of degree |m| ≥ 3 in [Gustafson et al 2010]. A certain energy-class instability for degree-1 solitons of (1-1) is shown in [Bejenaru and Tataru 2010], where it is also shown that global solutions always exist for small localized equivariant perturbations of degree-1 solitons.…”

The fractional Landau–Lifshitz–Gilbert equation and the heat flow of harmonic maps

“…The readers are referred to [1,2,13,18,19,26,29,33] for more details. Note that if |u 0 | = 1, it is easily to show that |u(t)| = 1 for all t ≥ 0.…”