Mathematical Sciences: Structural Stability for Parabolic Equations

9400233鲁该奖项支持抛物型偏微分方程组结构稳定性发展及相关问题的数学研究。动力系统的结构稳定性是动力系统理论中的基本问题之一。目前该领域研究人员面临的主要问题之一是局部非紧空间上的无限维动力系统。虽然已经取得了很大的进展，但那些由抛物型方程产生的系统只定义了半流，因为向后解可能不存在，相空间也不是局部紧的。本文将首先研究周期边界条件下标量方程的结构稳定性。特别是，将努力了解吸引子邻域的稳定性，并研究拟周期和几乎周期标量抛物型方程。这项工作涉及到倾斜产品流、圈数和弗洛奎特理论。还将开展含时抛物型方程的Floquet理论和抛物型方程的光滑共轭的其他研究。偏微分方程构成了对物理世界进行数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，对于充分描述现实的数学模型，当模型受到允许的小扰动时，解的定性属性不应显著改变。将这一观点形式化的尝试导致了结构稳定性的概念。***

9400233 Lu This award supports mathematical research on the development of structural stability for parabolic partial differential equations and related problems. The structural stability of dynamical systems is one of the fundamental problems in the theory of dynamical systems. One of the main problems currently facing researchers in the area concerns infinite dimensional dynamical systems on locally non-compact spaces. While much progress has been made, those systems generated by parabolic equations define only semiflows since backwards solutions may not exist and the phase spaces are not locally compact. This work will first focus on structural stability for scalar equations with periodic boundary conditions. In particular efforts will be made to understand stability in a neighborhood of an attractor and to study the quasiperiodic and almost periodic scalar parabolic equations. Involved in this work is the skew product flow, the lap number and Floquet theory. Other studies of Floquet theory for time dependent parabolic equations and smooth conjugacy for parabolic equations will also be carried out. Partial differential equations form a basis for mathematicalmodeling of the physical world. The role of mathematicalanalysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, for mathematical models to adequately describe reality, the qualitative properties of solutions should not change significantly when the model is subjected to small, allowable perturbations. An attempt to formalize this point of view leads to the concept of structural stability. ***