论文标题
大数据全局良好的和散射，用于$ \ mathbb {r}^2 \ times \ mathbb {t} $上的焦点非线性schrödinger方程
Large data global well-posedness and scattering for the focusing cubic nonlinear Schrödinger equation on $\mathbb{R}^2\times\mathbb{T}$
论文作者
论文摘要
我们考虑聚焦的立方非线性schrödinger方程\ begin {align} \ label {cnlss} i \ partial_t u+u+u+Δu= - | U |^2u \ quad \ quad \ quad \ quad \ text {on $ \ \ \ m} \ end {align}与3D欧几里得案例不同，\ eqref {cnlss}在waveGuide comploold $ \ mathbb {r}^2 \ times \ times \ times \ mathbb {t} $，因此分析变得更加庞大而挑战。我们使用聚焦立方NLS的2D欧几里得基态制定阈值，并表明位于阈值以下的\ eqref {cnlss}的解决方案是全局的，并且及时散射。证明依靠几个新的已建立的Gagliardo-Nirenberg不平等现象，其最佳常数是按照2d Euclidean地面状态制定的。还值得注意的是，一个有趣的事实是，全球适应性和散射的阈值不一致。据作者所知，本文还给出了将NLS集中在产品空间上的第一个大数据散射结果。
We consider the focusing cubic nonlinear Schrödinger equation \begin{align}\label{CNLSS} i\partial_t U+ΔU=-|U|^2U\quad\text{on $\mathbb{R}^2\times\mathbb{T}$}.\tag{3NLS} \end{align} Different from the 3D Euclidean case, the \eqref{CNLSS} is mass-critical and non-scale-invariant on the waveguide manifold $\mathbb{R}^2\times\mathbb{T}$, hence the underlying analysis becomes more subtle and challenging. We formulate thresholds using the 2D Euclidean ground state of the focusing cubic NLS and show that solutions of \eqref{CNLSS} lying below the thresholds are global and scattering in time. The proof relies on several new established Gagliardo-Nirenberg inequalities, whose best constants are formulated in term of the 2D Euclidean ground state. It is also worth noting the interesting fact that the thresholds for global well-posedness and scattering do not coincide. To the author's knowledge, this paper also gives the first large data scattering result for focusing NLS on product spaces.
