大能量波映象的几何重整化
Geometric renormalization of large energy wave maps
摘要：
近年来在理解具有小临界索伯列夫范数的波映射的存在性问题（特别是具有小能量的二维波映射）方面取得了很大进展;该理论的一个关键方面是重整化过程（几何库仑规范或微局域规范），它将非线性项转换为更接近半线性波方程的项。然而，这两个重整化过程遇到困难，如果能量的解决方案是大的。在这份报告中，我们提出了一个不同的重整化，基于调和映射热流，它适用于从二维到双曲空间的大能量波映射。我们还观察到一个有趣的“非集中”类型的估计，粗略地说，它断言，如果波图的能量集中在一个点，那么它变得渐近自相似。
英文摘要：
There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of ``non-concentration'' type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar.
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