Remarks on local theory for Schrödinger maps near harmonic maps
Remarks on local theory for Schrödinger maps near harmonic maps
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DOI：
10.2996/kmj/1594313555
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
Ikkei Shimizu
中科院分区：
文献类型：
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作者：
Ikkei Shimizu
本文考虑调和映射族附近的等变Schr“odinger映射的初值问题.本文对Gustafson，Kang和Tsai在[杜克数学杂志145（3）537- 583，2008]中关于局部适定性的证明提供了一些补充.我们还证明了调和映射附近的解在$C（I;\dot{H}^1（\mathbb{R}^2）\cap\dot{H}^2（\mathbb{R}^2））$中对于时间间隔$I$是唯一的。在证明中，我们证明了在低正则性条件下，修正的Schr“odinger映射方程的推导是合理的.
We consider the initial-value problem for the equivariant Schr\"odinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537--583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;\dot{H}^1(\mathbb{R}^2)\cap\dot{H}^2(\mathbb{R}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schr\"odinger map equation in low regularity settings without smallness of energy.