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Global Schrödinger maps in dimensions $d\ge 2$: small data in the critical Sobolev spaces. (English) Zbl 1233.35112

Two global results for the initial-value problem for the Schrödinger map equation, ${\partial }_{t}\phi =\phi ×{\Delta }\phi$, on ${ℝ}^{d}×ℝ$, $\phi \left(0\right)={\phi }_{0}$, $\phi :{ℝ}^{d}×ℝ\to {𝕊}^{2}$, $d\ge 2$, are proved. In order to formulate these results we have to introduce some notation. As usual, ${H}^{\sigma }$ denotes the Sobolev space on ${ℝ}^{d}$. For $Q\in {𝕊}^{2}$, ${H}_{Q}^{\sigma }=\left\{f:{ℝ}^{d}\to {ℝ}^{3\right)};|f\left(x\right)|=1,\text{a.e.},f-Q\in {H}^{\sigma }\right\}$, ${\parallel f\parallel }_{{H}_{Q}^{\sigma }}={\parallel f-Q\parallel }_{{H}^{\sigma }}$. The space ${H}_{Q}^{\infty }$ is the intersection of all spaces ${H}_{Q}^{\sigma }$ and for $f\in {H}^{\infty }$, ${\parallel f\parallel }_{{\stackrel{˙}{H}}^{\sigma }}$ is the homogeneous Sobolev norm.

The first theorem proved in the paper states that if $d\ge 2$ and $Q\in {𝕊}^{2}$, then there exist ${\epsilon }_{0}\left(d\right)>0$ and a constant $C>0$ such that for any ${\phi }_{0}\in {H}_{Q}^{\sigma }$ with $\parallel {\phi }_{0}{-Q\parallel }_{{\stackrel{˙}{H}}^{d/2}}\le {\epsilon }_{0}\left(d\right)$ there exists a unique solution $\phi ={S}_{Q}\left({\phi }_{0}\right)\in C\left(ℝ;{H}_{Q}^{\infty }\right)$ of the initial-value problem, ${\parallel \phi \left(t\right)-Q\parallel }_{{\stackrel{˙}{H}}^{d/2}}\le C{\parallel {\phi }_{0}-Q\parallel }_{{\stackrel{˙}{H}}^{d/2}},\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\in ℝ$ and for any $T\in \left[0,\infty \right)$ and $\sigma \in {ℤ}_{+}$, ${\text{sup}}_{t\in \left[-T,T\right]}{\parallel \phi \left(t\right)\parallel }_{{H}_{Q}^{\sigma }}\le C\left(\sigma ,T,\parallel {\phi }_{0}{\parallel }_{{H}_{Q}^{\sigma }}\right)$.

The second theorem provides uniform bounds (on $ℝ$) for ${\parallel \phi \left(t\right)\parallel }_{{H}_{Q}^{\sigma }}$ and asserts that the operator ${S}_{Q}$ admits continuous extensions ${S}_{Q}:{B}_{{\epsilon }_{0}\left(d,{\sigma }_{1}\right)}^{\sigma }\to C\left(ℝ;{\stackrel{˙}{H}}^{\sigma }\cap {\stackrel{˙}{H}}_{Q}^{d/2-1}\right)$ for any $\sigma \in \left[d/2,{\sigma }_{1}\right]$ and some ${\epsilon }_{0}\left(d,{\sigma }_{1}\right)\in \left(0,{\epsilon }_{0}\left(d\right)\right)$, where ${B}_{\epsilon }^{\sigma }=\left\{\phi \in {\stackrel{˙}{H}}_{Q}^{d/2-1}\cap {\stackrel{˙}{H}}^{\sigma }{;\parallel \phi -Q|}_{{\stackrel{˙}{H}}^{d/2}}\le \epsilon \right\}$.

The first theorem was proved in dimensions $d\ge 4$ in [I. Bejenaru, A. D. Ionescu, and 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 ${L}_{e}^{p,q}$ spaces and the caloric gauge.

##### MSC:
 35K45 Systems of second-order parabolic equations, initial value problems 35B65 Smoothness and regularity of solutions of PDE 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35Q41 Time-dependent Schrödinger equations, Dirac equations