【讲座题目】Stability and Instability on the De Gregorio Modificationof the Constantin-Lax-Majda model
【主 讲 人】酒全森 (首都师范大学)
【讲座时间】2025年6月18日 星期三下午4:00-5:00
【讲座地点】主楼C座636
【主讲人简介】 酒全森，首都师范大学数学科学学院教授，博士生导师。从事非线性偏微分方程以及流体方程的数学理论研究，在CMP 、ARMA、JFA、SIAM JMA等国内外权威核心（SCI）期刊上发表论文100余篇。于2003年获北京市科技新星项目，2013年获北京市“长城学者”人才项目，2024年获北京市教学名师。先后到香港中文大学、美国普林斯顿大学、美国Oklahoma州立大学、巴西Unicamp大学、法国萨瓦大学（Savoie University）等学术访问。曾主持国家自然科学基金4项，参加国家自然科学基金重点项目3项。
【讲座内容简介】The Constantin-Lax-Majda (CLM) model and the De Gregorio model which is a modification of the CLM model are well-known for their ability to emulate the behavior of the 3D Euler equations, particularly their potential to develop finite-time singularities. The stability properties of the De Gregorio model on the torus near the ground state $-\sin\theta$ have been well studied. However, the stability analysis near excited states $-\sin k\theta$ with $k\ge 2$ remains challenging.This paper focuses on analyzing the stability and instability of theDe Gregorio model on torus around the first excited state $-\sin 2\theta$.The linear and nonlinear instability are established for a broad class of initial data, while nonlinear stability is proved for another large class of initial data in this paper. Our analysisreveals thatsolution behavior to the De Gregorio model near excited states demonstrates different stability patterns depending on initial conditions. One of new ingredients in ourinstability analysis involves deriving a second-order ordinary differential equation (ODE) governing the Fourier coefficients of solutions and examining the spectral properties of a positive definite quadratic form emerging from this ODE. The approach of this paper would be applicable to other related models and problems.
