*(English)*Zbl 1122.35138

The author studies the Cauchy problem for Schrödinger maps from ${\mathbb{R}}^{d}\times {\mathbb{R}}^{+}$ $(d\ge 2)$ into a target manifold $N$ with a complex structure $J$. The Schrödinger map equation is given by

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 $\delta >0$

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.