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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 \(Q\in\mathbb{S}^{2}\), \(H_{Q}^{\sigma} = \{ f:\mathbb{R}^{d} \rightarrow \mathbb{R}^{3)}; | f(x) | =1, \text{a.e.}, f-Q \in H^{\sigma} \}\), \(\| f \|_{H_{Q}^{\sigma}} = \| f-Q \|_{H^{\sigma}}\). The space \(H_{Q}^{\infty}\) is the intersection of all spaces \(H_{Q}^{\sigma}\) and for \(f \in H^{\infty}\), \(\| f \| _{\dot H^{\sigma}}\) 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 \(\| \phi_{0}-Q \| _{\dot H ^{d/2}} \leq \varepsilon_{0}(d)\) there exists a unique solution \(\phi=S_{Q}(\phi_{0}) \in C(\mathbb{R};H_{Q}^{\infty})\) of the initial-value problem, \(\| \phi(t)-Q \| _{\dot H ^{d/2}} \leq C \| \phi_{0}-Q \| _{\dot H ^{d/2}}, \text{for all }t \in \mathbb{R}\) and for any \(T\in [0, \infty)\) and \(\sigma \in \mathbb{Z}_{+}\), \(\text{sup}_{t\in [-T,T]} \| \phi(t) \| _{H_{Q}^{\sigma}} \leq C(\sigma, T, \| \phi_{0} \| _{H_{Q}^{\sigma}})\).
The second theorem provides uniform bounds (on \(\mathbb{R}\)) for \(\| \phi(t) \| _{H_{Q}^{\sigma}}\) 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 \(B^{\sigma}_{\varepsilon}= \{ \phi \in \dot H_{Q}^{d/2-1} \cap \dot H^{\sigma}; \| \phi-Q | _{\dot H ^{d/2}} \leq \varepsilon \}\).
The first theorem was proved in dimensions \(d\geq4\) 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 \(L^{p,q}_{e}\) spaces and the caloric gauge.

MSC:

35K45 Initial value problems for second-order parabolic systems
35B65 Smoothness and regularity of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q41 Time-dependent Schrödinger equations and Dirac equations

Citations:

Zbl 1152.35049
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References:

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