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An approximation scheme for Schrödinger maps. (English) Zbl 1122.35138

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

${\partial }_{t}u=J\left(u\right){D}_{k}{\partial }_{k}u,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $D$ is the covariant derivative along the curve $u$. A Schrödinger map is a function $u:\left[0,T\right]×{ℝ}^{d}\to N$ 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 $\delta >0$

${\delta }^{2}{D}_{t}{\partial }_{t}{u}^{\delta }-J\left({u}^{\delta }\right){\partial }_{t}{u}^{\delta }-{D}_{m}{\partial }_{m}{u}^{\delta }=0·\phantom{\rule{2.em}{0ex}}\left(2\right)$

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 $\left[0,{T}_{\delta }\right]$. 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 $\left[0,T\right]$, 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-like (nonlinear Schrödinger) equations 35A35 Theoretical approximation to solutions of PDE 35G25 Initial value problems for nonlinear higher-order PDE