Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions (Q2925681)

In the very interesting memoir under review, the authors consider the Schrödinger map flow equation in \(2+1\) dimensions, with values into the 2-sphere \(\mathbb{S}^2,\) whose initial data are close to a fixed point of the flow, that is, a harmonic map. This equation admits a lowest energy steady state \(Q\) -- the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that \(Q\) is unstable in the energy space \(\dot H^1\) and, proving this, it is also shown that within the equivariant class \(Q\) is stable in a topology \(X\) slightly stronger than \(\dot H^1\) but weaker than \(H^1\). Moreover, this stability result turns out to be close to optimal.