Mathematical Sciences: Structural Stability and Floquet Theory for Parabolic Equations

本研究的主要目的是发展抛物型偏微分方程解的局部和全局结构稳定性理论和含时线性抛物型方程的Floquet理论。结构稳定性最早是在常微分方程组的研究中被引入的，特别是对于动力系统。动力系统的许多概念已经适用于无限维系统，特别是定义非紧空间上的动力系统的偏微分方程组。然而，那些由抛物方程产生的系统是不可逆的。这一特点给有限维动力系统的研究带来了前所未有的困难，必须发展新的方法来理解这些系统的本质。标量反应扩散方程的一种理论是最近得到的。具体的分析将考虑特定类型的方程，如Cahn-Hilliard方程和相场系统。其他工作将研究时间周期抛物型方程周期解附近流动的结构稳定性。本着同样的精神，我们将努力从有限周期系统到类似类型的抛物型偏微分方程组，将Floquet理论(技术太强，无法在这里描述)继续下去。

The main goal of this research is the development of a local and global structural stability theory for parabolic partial differential equations and a Floquet theory for time dependent linear parabolic equations. Structural stability was first introduced in the study of ordinary differential equations, in particular for dynamical systems. Many concepts of dynamical systems have been adapted to infinite dimensional systems, especially to partial differential equations which define dynamical systems on non-compact spaces. However, those systems generated by parabolic equations are not reversible. This characteristic creates difficulties not encountered in the study of finite dimensional dynamical systems and new methods must be developed to understand the nature of these systems. A theory for scalar reaction-diffusion equations has recently been obtained. Specific classes of equations such as the Cahn-Hilliard Equation and Phase-Field System will be considered for specific analysis. Other work will study structural stability of flows near periodic solutions for time-periodic parabolic equations. In the same spirit, efforts will be made to carry over the Floquet theory (too technical to describe here) from finite periodic systems to parabolic partial differential equations of similar type.