论文标题
在使用超临界数据的临界立方波方程的几乎确定的散射上
On the almost sure scattering for the energy-critical cubic wave equation with supercritical data
论文作者
论文摘要
在本文中，我们研究了$ \ mathbb {r}^4 $与超临界数据的偶然偏置非线性波方程。我们证明，在$ h^s（\ mathbb {r}^4）中的随机初始数据的散射几乎可以肯定，\ times h^{s-1}（\ mathbb {r}^4）$带有$ \ frac {5} {6} {6} <s <s <1 $。证明依赖于带有随机数据的波方程线性流动的新概率估计值，其中随机化基于频率空间中的单位尺度分解，角变量中的分解以及物理空间的单位尺度分解。特别是，我们表明，使用随机数据的线性波方程的解决方案几乎肯定属于$ l^1_t l^\ infty_x $。
In this article we study the defocusing energy-critical nonlinear wave equation on $\mathbb{R}^4$ with scaling supercritical data. We prove almost sure scattering for randomized initial data in $H^s(\mathbb{R}^4) \times H^{s-1}(\mathbb{R}^4)$ with $\frac{5}{6} < s < 1$. The proof relies on new probabilistic estimates for the linear flow of the wave equation with randomized data, where the randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition of physical space. In particular, we show that the solution to the linear wave equation with randomized data almost surely belongs to $L^1_t L^\infty_x$.
