论文标题
在随机扰动下，小基接地状态的渐近稳定性
Asymptotic stability of small ground states for NLS under random perturbations
论文作者
论文摘要
我们认为欧几里得空间上的CupicSchrödinger方程式受到短期潜在$ V $的干扰。 $-Δ+V $的负简单特征值的存在导致小型和局部非线性基础状态的曲线，这些曲线产生了某些已知在能量空间中渐近稳定的时间周期溶液。我们研究这些相干状态在粗糙的扰动下的持久性。我们将在能量空间下方构建一组大尺度缩放质的解决方案，这些解决方案显示出一些渐近稳定性行为。主要困难是需要在调制方程中处理本地化和分散项的相互作用。为此，我们使用关键加权策略将基于$ u^p，v^q $（控制高阶项）（控制较高订单术语）与某些局部能量衰减估计值（控制较低订单项）的关键空间中的概率非线性估计结合在一起。我们还在扰动的设置Bényi，OH和Pocovnicu的分析中对概率的全球范围良好和散射，以供小型超临界初始数据进行分析。我们使用扭曲的傅立叶变换和半经典函数积分来概括概率和双线性strichartz估计。
We consider the cubic Schrödinger equation on the euclidean space perturbed by a short-range potential $V$. The presence of a negative simple eigenvalue for $-Δ+V$ gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behavior. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on $U^p, V^q$ (controlling higher order terms) with some local energy decay estimates (controlling lower order terms). We also revisit in the perturbed setting the analysis of Bényi, Oh and Pocovnicu on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorted Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.
