Mathematical Sciences: Structural Stability for Parabolic Equations

小行星9400233 该奖项支持抛物型偏微分方程及相关问题的结构稳定性发展的数学研究。 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。 目前研究人员在该领域面临的主要问题之一涉及局部非紧空间上的无穷维动力系统。 虽然已经取得了很大的进展，这些系统所产生的抛物型方程只定义半流，因为向后的解决方案可能不存在，相空间不是局部紧的。 本文将首先研究具有周期边界条件的标量方程的结构稳定性。 特别是将努力了解吸引子的邻域稳定性，并研究拟周期和概周期标量抛物方程。 本文涉及的内容包括斜交产品流、圈数和Floquet理论。 其他研究的Floquet理论的时间依赖抛物方程和光滑共轭抛物方程也将进行。 偏微分方程是物理世界数学模型的基础。 数学分析的作用与其说是建立方程，不如说是提供有关解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，对于充分描述现实的数学模型，当模型受到小的、允许的扰动时，解的定性性质不应发生显著变化。 试图将这一观点形式化，就引出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. ***