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。特别有趣的是研究各种类型的偏微分方程解的水平集的大小、最大消失阶以及解的增长性质。该方法利用椭圆正则性理论，简化和推广了H.Donnelly，C.Fefferman和F.-H.Lin关于二阶偏微分方程解的几何性质的结果。基于Lions和Magenes椭圆迭代定理的PI方法推广到一类具有解析系数的线性和非线性椭圆型和抛物型方程。PI提出了一些仍未解决的问题，特别是关于水平集的大小和具有非解析系数的非线性发展方程和方程的最大消失阶的精确界。非线性发展方程，如Navier-Stokes系统，与许多数学物理问题有关。数值模型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，即解呈现复杂的时间和空间振荡。本项目致力于解决这一问题，并旨在获得解决方案的诸如其振荡行为和其生长特性等的亲密信息。关于这些问题的信息，这是本项目的主要目的，将有助于更好地理解在流体力学和相关领域的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.