论文标题
$ d> 6 $中的能量关键NLS几乎确定散射
Almost Sure Scattering of the Energy Critical NLS in $d>6$
论文作者
论文摘要
我们研究了具有随机初始数据$ d> 6 $的随机初始数据的能源非线性schrödinger方程。我们证明，Cauchy问题几乎肯定在全球范围内都有良好的功能，并在$ h^s中（\ Mathbb {r}^d）$中的随机超临界初始数据进行散射。 $ s> \ max \ {\ frac {4d-1} {3（2d-1）}，\ frac {d^2+6d-4} {（2dd-1）（d+2）} \} $。随机分析基于物理空间，频率空间和角变量中数据的分解。这扩展了Spitz在维度4中的先前已知结果。对高维度的概括的主要困难是非线性的不平滑度。
We study the energy-critical nonlinear Schrödinger equation with randomised initial data in dimensions $d>6$. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $H^s(\mathbb{R}^d)$ whenever $s>\max\{\frac{4d-1}{3(2d-1)},\frac{d^2+6d-4}{(2d-1)(d+2)}\}$. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results of Spitz in dimension 4. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.
