Mathematical Sciences: Geometric Properties of Solutions of Partial Differential Equations

摘要：本课题研究偏微分方程解的某些几何性质。特别感兴趣的是研究水平集的大小，消失的最大阶数，以及各种类型的偏微分方程解的增长性质。所提出的方法是利用椭圆正则性理论来寻找H. Donnelly， C. Fefferman和F.-H.的结果的简化和扩展。二阶偏微分方程解的几何性质。PI方法基于Lions和Magenes的椭圆迭代定理，将其推广到一类具有解析系数的椭圆型和抛物型线性和非线性方程。PI提出了一些仍然没有解决的问题，特别是包括水平集大小的尖锐界限和非线性演化方程和非解析系数方程的最大消失阶。非线性演化方程，如Navier-Stokes系统，在许多数学物理问题中出现。数值和实验模型支持这样的理论，即这些方程产生解的混沌行为，即解表现出复杂的时间和空间振荡。本项目解决了这一问题，旨在获得解的振荡行为和生长特性等私密信息。关于这些问题的信息是本项目的主要目的，将有助于更好地理解流体力学和相关领域许多问题中出现的振荡现象。

Abstract This project concerns certain geometric properties of solutions of partial differential equations. Of particular interest are a study of the size of level sets, the maximal order of vanishing, and growth properties of solutions of various types of partial differential equations. The proposed method is a use of elliptic regularity theory to find simplifications and extensions of the results of H. Donnelly, C. Fefferman, and F.-H. Lin concerning geometric properties of solutions of second order partial differential equations. The PI's method, based on the theorem on elliptic iterates of Lions and Magenes, leads to extensions to a wide class of linear and nonlinear equations of elliptic and parabolic type with analytic coefficients. The PI proposes to address some questions which still remain unresolved, including, in particular, sharp bounds on the size of level sets and the maximal order of vanishing for nonlinear evolution equations and equations with non-analytic coefficients. Nonlinear evolution equations, such as the Navier-Stokes system, arise in in connection with numerous problems of mathematical physics. Numerical and experimental models support the theory that such equations produce chaotic behavior of solutions, i.e., solutions which exhibit complex temporal and spatial oscillations. This project addresses this issues and aims to obtain such intimate information of solutions as their oscillatory behavior and their growth properties. Information on these issues, which is the main purpose of this project, would lead to better understanding of oscillatory phenomena arising in many problems of fluid mechanics and related fields.