系列报告题目：Nonlinear Stability of Periodic Patterns

报告人：吴启亮教授，美国俄亥俄大学

报告时间、地点：

· 2024年12月16日（周一）9:00-11:00，20-306

· 2024年12月19日（周四）9:00-11:00，21-427

· 2024年12月20日（周五）14:30-16:30，20-306

报告摘要：

Part I: Existence of Periodic Patterns and Defects

Part II: Spectral Analysis of Periodic Patterns

Part III: Nonlinear Stability of Periodic Patterns

Abstract: The existence and stability of pattern solutions is a fundamental question in the study of pattern formation. 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 spectrally stable patterns, various techniques, including renormalization, mode filter, phase modulation and pointwise Green’s function estimates, are introduced to prove 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 spectral 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 $t^{-1}$ diffusive decay rate, small perturbations of zigzag stable roll solutions decay with slower algebraic rates ($t^{-3/4}$ for the 2D plane; $t^{-1/4}$ 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 irrelevancy 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。

邀请人：现代分析及其应用数学研究所、动力系统与非线性分析研究所