短期课程名称：Dynamics analysis in chemotaxis and fluid mechanics

授课人：周涛（新加坡国立大学）

时间、地点：12月16日（周二），下午4:00-6:00，5106教室

                    12月18日（周四），上午10:00-12:00，5202教室

摘   要：This mini-course provides a brief introduction to partial differential equations arising from chemotaxis and fluid dynamics.

Part I. The first part is devoted to singularity formation in PDEs describing chemotaxis. We will focus on 3D Keller–Segel equation, in particular in regimes where the nonlocal effects cannot be ignored. The discussion will center on two problems:

1. Nonradial stability of self-similar blowup solutions to the 3D Keller–Segel equation.

2. Sharp blowup results for the 3D Keller–Segel equation with logistic damping.

For Problem (1), the main ingredient is the mode-stability analysis of the linearized operator. Besides a quantitative perturbative analysis for higher spherical modes, we adapt in the first spherical mode the wave-operator method of Li–Wei–Zhang (2020) from fluid stability in order to localize the operator and simultaneously remove the known unstable mode.

For Problem (2), we step outside the usual mindset of constructing a self-similar solution satisfying the exact scaling invariance of the original model, which is highly challenging. Instead, we construct a self-similar blowup solution to a related aggregation equation with subcritical scaling as an approximate profile. Magically, in the resulting phase-portrait analysis, the threshold \mu = 1/3 naturally emerges in a sharp way.

Part I is based on joint work with Zexing Li, Jiaqi Liu, and Yixuan Wang.

Part II. The second part concerns the stability of certain steady states and the creation of small scale for the two-dimensional Euler equation on T^2 and R^2. We will first introduce several important steady solutions on  T^2 and R^2, and briefly outline techniques to study their stability. In particular, we identify orbitally stable steady states that generate a saddle point structure (e.g. w(x,y) = \sin x + \sin y on T^2 and the Lamb–Chaplygin dipole on R^2), which exhibit strong stretching effects at some points and are natural candidates for infinite-time growth of the vorticity gradient. Building on these observations, we prove the existence of solutions with superlinear growth of the vorticity gradient on both T^2 and R^2.

Part II is based on joint work with In-Jee Jeong and Yao Yao.

授课人简介：周涛，新加坡国立大学数学系在读博士，研究方向主要集中在来源于生物数学和流体力学的偏微分方程的动力学行为，主要成果发表在CMP等重要期刊。