Spectral methods for dispersive equations (A09)

谱理论提供了一个统一的方法，以各种各样的微分算子出现在色散方程。在这个项目中，我们直接通过自伴算子的谱数据，高斯边界和由泛函微积分提供的振荡积分表示来研究谱局部色散估计。此外，我们使用随机化的谱分解的运营商，以改善经典的Sobolev嵌入，针对大的初始数据集的适定性。最后，我们将Keel-Tao方法应用于随机环境中，并以非线性薛定谔方程等随机色散方程的新适定性结果为目标。

Spectral theory provides a unified approach to a large variety of differential operators appearing in dispersive equations. In this project we investigate spectrally localized dispersive estimates directly via the spectral data of selfadjoint operators, Gaussian bounds, and oscillatory integral representations provided by functional calculus. Moreover, we use randomizations of spectral decompositions of operators to improve classical Sobolev embeddings, aiming at well-posedness for large sets of initial data. Finally, we apply the Keel-Tao method of proving Strichartz estimates to the stochastic setting and with the goal of new well-posedness results for stochastic dispersive equations like the nonlinear Schrödinger equation.