报告人：吴奕飞（南京师范大学数学科学学院院长、教授、博士生导师）
时间：1月14日15:10-15:50
地点：36-507
摘要：In this talk, we consider the cubic nonlinear Schrodinger equation (NLS) with a spatially rough potential, a key model for nonlinear Anderson localization. Given its importance in simulations, the previous numerical and the analytical study cover only $L^2$-potential case. We go beyond the limit in this paper for optimal computations, finding and covering the roughest possible potentials within the well-posedness. Our result comprises three main parts:
(1) Sharp PDE theory: We establish new global well-posedness and ill-posedness thresholds in Sobolev and Fourier–Besov-type spaces for the NLS with potentials as singular as the Dirac function. We quantify how the regularity of the solution depends explicitly and optimally on that of the potential.
(2) Optimal numerical analysis: Based on the delicate PDE theory, we design a new low-regularity integrator tailored to rough potentials, for which we prove convergence rates with sharp regularity dependence.
(3) Numerical verification: Our simulations not only confirm the predicted regularity theory and error bounds on scheme but also demonstrate superior performance over traditional schemes in the rough regime.
报告人简介：吴奕飞，南京师范大学数学科学学院院长、教授、博士生导师。主要从事偏微分方程理论及数值分析、调和分析等方向的交叉研究，在非线性Schrödinger方程、KdV方程等整体适定性和低正则算法构造方面做出一系列研究成果，科研论文发表在J. Eur. Math. Soc、Found. Comput. Math.、Comm. Math. Phy.、Adv. Math.、Anal. PDE、Numer. Math、Math. Comp.等国际刊物上，2023年入选国家级领军人才.
