Global regularity of wave maps, I: small critical Sobolev norm in high dimension
Global regularity of wave maps, I: small critical Sobolev norm in high dimension
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DOI：
10.1155/s1073792801000150
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发表时间：
2000-10
影响因子：
1
通讯作者：
T. Tao
中科院分区：
文献类型：
--
作者：
T. Tao
我们证明了从Minkowski空间$R^{1+n}$到球面的波映射是全局光滑的，如果初始数据是光滑的，并且在高维情形下在临界Sobolev空间$\dot H^{n/2}$中具有小范数。一个主要的困难，不存在于早期的结果，是，$\dot H^{n/2}$范数几乎无法控制$L^\infty$，潜在地导致对数发散的非线性，然而，这可以克服使用坐标系适应波图近似平行运输。在本文的续集中，我们解决了更有趣的二维情况下，这是能量临界。
We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical.