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Zbl 1122.35138
McGahagan, Helena
An approximation scheme for Schrödinger maps.
(English)
[J] Commun. Partial Differ. Equations 32, No. 3, 375-400 (2007). ISSN 0360-5302; ISSN 1532-4133/e

The author studies the Cauchy problem for Schrödinger maps from $\Bbb R^d\times \Bbb R^+$ $(d \ge 2)$ into a target manifold $N$ with a complex structure $J$. The Schrödinger map equation is given by $$\partial _t u = J(u)D_k \partial _k u, \tag 1$$ where $D$ is the covariant derivative along the curve $u$. A Schrödinger map is a function $u:[0,T]\times \Bbb R^d \to N$ that solves the equation (1). To show that solutions of the Cauchy problem for the Schrödinger map equation (1) exist, the author first studies the following approximate equation for any $\delta > 0$ $$\delta ^2D_t \partial_t u^\delta - J(u^\delta )\partial_t u^\delta - D_m \partial_m u^\delta = 0. \tag 2$$ The equation (2) is a wave map, for which general existence theory is known. For appropriate initial data there is a sequence of local solutions $u^\delta$ of equation (2) that exist on the time intervals $[0,T_\delta]$. The limit of these approximate solutions as $\delta \to 0$ solves the Schrödinger map problem. Energy estimates which bound the norms of solutions $u^\delta$ imply that $T_\delta$ is independent of $\delta$. Then, for some fixed $T > 0$, the time interval of existence for each solution $u^\delta$ is $[0,T]$, and their limit $u$ exist on this same interval. By a standard convergence argument it is proven that $u$ satisfies the Schrödinger map equation (1). The uniqueness of the Schrödinger map is also shown.
MSC 2000:
*35Q55 NLS-like (nonlinear Schroedinger) equations
35A35 Theoretical approximation to solutions of PDE
35G25 Initial value problems for nonlinear higher-order PDE

Keywords: nonlinear Schrödinger equations; initial value problem; local well-posedness

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