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On the high–low method for NLS on the hyperbolic space
Journal of Mathematical Physics ( IF 1.3 ) Pub Date : 2020-08-01 , DOI: 10.1063/5.0012061
Gigliola Staffilani 1 , Xueying Yu 1
Affiliation  

In this paper, we first prove that the cubic, defocusing nonlinear Schr\"odinger equation on the two dimensional hyperbolic space with radial initial data in $H^s(\mathbb{H}^2)$ is globally well-posed and scatters when $s > \frac{3}{4}$. Then we extend the result to nonlineraities of order $p>3$. The result is proved by extending the high-low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by the first author and Ionescu.

中文翻译:

双曲空间上NLS的高低法

在本文中,我们首先证明了二维双曲空间上具有 $H^s(\mathbb{H}^2)$ 中径向初始数据的三次散焦非线性 Schr\"odinger 方程是全局适定的并且散布当 $s > \frac{3}{4}$. 然后我们将结果扩展到 $p>3$ 阶的非线性。通过在双曲线设置中扩展 Bourgain 的高低方法并使用由第一作者和 Ionescu 证明的 Morawetz 类型估计。
更新日期:2020-08-01
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