课程题目：Nonlinear Stability of Periodic Patterns
时 间：2024-11-25 (星期一) 14:00-17:00
2024-11-26 (星期二) 09:00-12:00
2024-11-27 (星期三) 09:00-12:00
2024-11-28 (星期四) 14:00-17:00
2024-11-29 (星期五) 12:00-17:00
地 点：理工北楼601
主 办：数学与统计学院、分析数学及应用教育部重点实验室、福建省分析数学及应用重点实验室、统计学与人工智能福建省高校重点实验室、福建省应用数学中心（福建师范大学）
参加对象：感兴趣的老师和研究生
课程摘要：Pattern formation, arising ubiquitously in biological, chemical and physical systems, has been an active research area for over a century. Among others, the existence and stability of patterns and their defects in pattern forming systems have been always one of the fundamental topics. There are two types of stability: The Lyapunov stability refers to the stability of solutions with respect to initial perturbations as time goes to infinity the structural stability concerns the persistence of solutions with respect to perturbations to the original system. We study the structural stability of patterns with an emphasis on the formation mechanism of their defects.
In this talk series, we firstly introduce the existence of spatially periodic patterns and some defects in the Swift-Hohenberg equation a prototypical pattern forming system, showcasing the application of basic tools such as the Lyapunov-Schmidt reduction and the center manifold theorem in the study of pattern formation. We then perform a systematic spectral analysis of these small-amplitude spatially periodic patterns via Bloch-Fourier analysis. For spectral stable patterns, various technigues. including renormalization, mode filter, phase modulation and pointwise Green's function estimates, are introduced to proof their nonlinear stability. At last, we present our recent results on the case when the spectrum of the linearization is diffusively stable with high-order spectra degeneracy at the origin. Roll solutions at the zigzag boundary of the Swift-Hohenberg equation are shown to be nonlinearly stable, serving as examples that linear decays weaker than the classical diffusive decay, together with quadratic nonlinearity, still give nonlinear stability of spatially periodic patterns. The study is conducted on two physical domains: the 2D plane and the infinite 2D torus. Linear analysis reveals that, instead of the classical $t~f-1$diffusive decay rate, small perturbations of zigzag stable roll solutions decay with slower algebraic rates ($t~{-3/4}$for the 2D plane; $t^y-1/4}$ for the infinite 2D torus) due to the high-order degeneracy of the translational mode at the origin in the Bloch-Fourier spaces. The nonlinear stability proofs are based on decompositions of the neutral translational mode and the faster decaying modes, and fixed-point arguments, demonstrating the irelevancy of the nonlinear terms.
报告人简介：吴启亮，美国俄亥俄大学教授，研究领域为非线性动力系统微分方程，生物数学。本科毕业于中国科技大学，2013年于美国明尼苏达大学获博士学位，后于密歇根州立大学作博士后研究。其研究获美国国家自然科学基金资助。在JDE，JMPA，JMB，PRSE等国际权威杂志发表论文数十篇。