Conditional global regularity of Schrödinger maps: Subthreshold dispersed energy

Abstract

We consider the Schrödinger map initial value problem

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ {\partial }_t\phi =\phi \times \Delta \phi ,\hfill \\ \hfill \\ \hfill \\ \phi \left(x,0\right)={\phi }_0\left(x\right),\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

with ${\phi }_0:{ℝ}^2\to {\S }^2\hookrightarrow {ℝ}^3$ a smooth $H_Q^{\infty }$ map from the Euclidean space ${ℝ}^2$ to the sphere ${\S }^2$ with subthreshold ($<4\pi$) energy. Assuming an a priori $L^4$ boundedness condition on the solution $\phi$, we prove that the Schrödinger map system admits a unique global smooth solution $\phi \in C\left(ℝ\to H_Q^{\infty }\right)$ provided that the initial data ${\phi }_0$ is sufficiently energy-dispersed, i.e., sufficiently small in the critical Besov space ${Ḃ}_{2,\infty }^1$. Also shown are global-in-time bounds on certain Sobolev norms of $\phi$. Toward these ends we establish improved local smoothing and bilinear Strichartz estimates, adapting the Planchon–Vega approach to such estimates to the nonlinear setting of Schrödinger maps.