Conditional global regularity of Schroedinger maps: sub-threshold dispersed energy

Details Conditional global regularity of Schroedinger maps: sub-threshold dispersed energy Conditional global regularity of Schroedinger maps: sub-threshold dispersed energy

2010-12-18

arxiv.org/abs/1012.4048v1

Mathematics

Analysis of PDEs

Scientific paper

We consider the Schr\"odinger map initial value problem {equation*} {cases} \partial_t \phi &= \quad \phi \times \Delta \phi \phi(x, 0) &= \quad \phi_0(x), {cases} \label{SMa} {equation*} with $\phi_0 : \mathbf{R}^2 \to \mathbf{S}^2 \hookrightarrow \mathbf{R}^3$ a smooth $H_Q^\infty$ map from the Euclidean space $\mathbf{R}^2$ to the sphere $\mathbf{S}^2$. Our main result is a step toward verifying the threshold conjecture. In particular, given energy-dispersed data $\phi_0$ with sub-threshold energy, we prove that the Schr\"odinger map system admits a unique global smooth solution $\phi \in C(\mathbf{R} \to H_Q^\infty)$ provided that the gradient $\partial_x \phi$ respects an a priori $L^4$ boundedness condition. Also shown is the absence of weak turbulence. Toward these ends we establish improved local smoothing and bilinear estimates, introducing a more covariant, non-perturbative, physical space-based approach of independent interest. Our work constitutes the first application over the whole sub-threshold energy range of the sub-threshold caloric gauge developed in \cite{Sm09}.

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