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Global Schrödinger maps in dimensions $d\geq 2$: small data in the critical Sobolev spaces. (English) Zbl 1233.35112
Two global results for the initial-value problem for the Schrödinger map equation, $\partial_{t}\phi=\phi\times \Delta\phi$, on $\mathbb{R}^{d} \times \mathbb{R}$, $\phi(0)=\phi_{0}$, $\phi:\mathbb{R}^{d} \times \mathbb{R} \rightarrow \mathbb{S}^{2}$, $d\geq2$, are proved. In order to formulate these results we have to introduce some notation. As usual, $H^{\sigma}$ denotes the Sobolev space on $\mathbb{R}^d$. For $Q\in\mathbb{S}^{2}$, $H_{Q}^{\sigma} = \{ f:\mathbb{R}^{d} \rightarrow \mathbb{R}^{3)}; \vert f(x) \vert =1, \text{a.e.}, f-Q \in H^{\sigma} \}$, $\Vert f \Vert_{H_{Q}^{\sigma}} = \Vert f-Q \Vert_{H^{\sigma}}$. The space $H_{Q}^{\infty}$ is the intersection of all spaces $H_{Q}^{\sigma}$ and for $f \in H^{\infty}$, $\Vert f \Vert _{\dot H^{\sigma}}$ is the homogeneous Sobolev norm. The first theorem proved in the paper states that if $d\geq2$ and $Q\in \mathbb{S}^{2}$, then there exist $\varepsilon_{0}(d)>0$ and a constant $C>0$ such that for any $\phi_{0} \in H_{Q}^{\sigma}$ with $\Vert \phi_{0}-Q \Vert _{\dot H ^{d/2}} \leq \varepsilon_{0}(d)$ there exists a unique solution $\phi=S_{Q}(\phi_{0}) \in C(\mathbb{R};H_{Q}^{\infty})$ of the initial-value problem, $\Vert \phi(t)-Q \Vert _{\dot H ^{d/2}} \leq C \Vert \phi_{0}-Q \Vert _{\dot H ^{d/2}}, \text{for all }t \in \mathbb{R}$ and for any $T\in [0, \infty)$ and $\sigma \in \mathbb{Z}_{+}$, $\text{sup}_{t\in [-T,T]} \Vert \phi(t) \Vert _{H_{Q}^{\sigma}} \leq C(\sigma, T, \Vert \phi_{0} \Vert _{H_{Q}^{\sigma}})$. The second theorem provides uniform bounds (on $\mathbb{R}$) for $\Vert \phi(t) \Vert _{H_{Q}^{\sigma}}$ and asserts that the operator $S_{Q}$ admits continuous extensions $S_{Q}:B^{\sigma}_{\varepsilon_{0}(d, \sigma_{1})} \rightarrow C(\mathbb{R};\dot H^{\sigma} \cap \dot H_{Q}^{d/2-1})$ for any $\sigma \in [d/2, \sigma_{1}]$ and some $\varepsilon_{0}(d, \sigma_{1}) \in (0, \varepsilon_{0}(d))$, where $B^{\sigma}_{\varepsilon}= \{ \phi \in \dot H_{Q}^{d/2-1} \cap \dot H^{\sigma}; \Vert \phi-Q \vert _{\dot H ^{d/2}} \leq \varepsilon \}$. The first theorem was proved in dimensions $d\geq4$ in [{\it I. Bejenaru, A. D. Ionescu}, and {\it C. E. Kenig}, Adv. Math. 215, No. 1, 263--291 (2007; Zbl 1152.35049)]. In order to overcome the difficulties which appear, especially in the case $d=2$, the authors of the paper under review use a sum of Galilei transforms of the lateral $L^{p,q}_{e}$ spaces and the caloric gauge.

35K45Systems of second-order parabolic equations, initial value problems
35B65Smoothness and regularity of solutions of PDE
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q41Time-dependent Schrödinger equations, Dirac equations
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