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An approximation scheme for Schrödinger maps. (English) Zbl 1122.35138
The author studies the Cauchy problem for Schrödinger maps from $$\mathbb R^d\times \mathbb R^+$$ $$(d \geq 2)$$ 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 $$u:[0,T]\times \mathbb R^d \to N$$ 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:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35A35 Theoretical approximation in context of PDEs 35G25 Initial value problems for nonlinear higher-order PDEs
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