Schrodinger Flow Near Harmonic Maps

TL;DR

We address the Schrödinger flow $u_t = u\times \Delta u$ from $\mathbb{R}^2$ to $\mathbb{S}^2$ in the $m$-equivariant, energy-critical setting, focusing on initial data with energy near the harmonic-map minimum $4\pi|m|$. The authors construct a nonlinear modulation projection onto the harmonic-map orbit $\mathcal{O}_m$ and show orbital stability: solutions remain near $\mathcal{O}_m$ up to the maximal existence time. They establish a sharp blow-up criterion: finite-time blow-up occurs if and only if the length-scale $s(t)$ of the nearest harmonic map collapses to zero, with the opposite scenario yielding global behavior. The analysis combines a Hasimoto-type reduction, modulation equations, and Strichartz estimates to convert energy excess into precise control of the perturbation.

Abstract

For the Schr\"odinger flow from $R^2 \times R^+$ to the 2-sphere $S^2$, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to zero.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• Lemma 4.1