Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2

TL;DR

This paper investigates the long-term behavior of solutions to certain geometric flow equations on R^2, establishing stability results for higher degrees and revealing complex dynamics like blow-up and oscillation at lower degrees.

Contribution

It extends asymptotic stability results to degree m=3 in Landau-Lifshitz equations and introduces new analytical tools to handle slow decay and complex dynamics.

Findings

Proved convergence to harmonic maps for m ≥ 3

Identified potential for blow-up and oscillation at m=2 under symmetry restrictions

Developed a new approach involving a normal form and Strichartz estimates

Abstract

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work 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 Schroedinger 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".