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Zbl 1152.35049
Bejenaru, I.; Ionescu, A.D.; Kenig, C.E.
Global existence and uniqueness of Schrödinger maps in dimensions $d\geq 4$.
(English)
[J] Adv. Math. 215, No. 1, 263-291 (2007). ISSN 0001-8708

For $\sigma\geq 0$ and $n\in \{1,2,\dots \}$ let $H^\sigma=H^\sigma({\Bbb R}^d, {\Bbb C}^n)$ denote the Banach space of ${\Bbb C}^n$-valued Sobolev functions on ${\Bbb R}^d$. For $\sigma\geq 0$ and $Q=(Q_1,Q_2,Q_3)\in {\Bbb S}^2$ define complete metric space $H_Q^\sigma = H_Q^\sigma({\Bbb R}^d;{\Bbb S}^2\hookrightarrow {\Bbb R}^3) = \{f:{\Bbb R}^d\to{\Bbb R}^3; \vert f(x)\vert \equiv 1, f-Q\in H^\sigma \}$ with induced distance $d_Q^\sigma(f,g) =\Vert f-g\Vert _{H^\sigma}$, and $H^\infty_Q=\bigcap_{\sigma\in{\Bbb Z}_+} H_Q^\sigma$. Let $s:{\Bbb R}^d\times{\Bbb R}\to {\Bbb S}^2\hookrightarrow {\Bbb R}^3$ is a continuous function. The authors consider the Schrödinger map initial-value problem $$\cases \partial s = s\times \Delta_x s,\quad \text{ on}\quad {\Bbb R}^d \times {\Bbb R} \cr s(0)=s_0.\endcases$$ \par It is proved that in dimensions $d\geq 4$ this problem admits a unique global (in time) solution $s\in C({\Bbb R}:H_Q^\infty)$, provided that $s_0\in H_Q^\infty$ and $\Vert s_0-Q\Vert _{H^{d/2}}\ll 1$, where $Q\in {\Bbb S}^2$.
[Michael Perelmuter (Ky{\"\i}v)]
MSC 2000:
*35K55 Nonlinear parabolic equations
35K15 Second order parabolic equations, initial value problems
35K60 (Nonlinear) BVP for (non)linear parabolic equations
46E35 Sobolev spaces and generalizations
47H99 Nonlinear operators

Keywords: Schrödinger maps; modified Schrödinger maps; orthonormal frames; a priori estimates

Cited in: Zbl 1233.35112

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