Numerical computation and qualitative properties of nonlinear Partial Differential Equations

获奖摘要0504720，J Szeftel，普林斯顿大学题目：非线性偏微分方程的数值计算与定性性质。第一个问题是关于非线性偏微分方程吸收边界条件的设计。主要目标是获得流体动力学等系统的吸收边界条件。这项工作将包括数值计算以及先进的微局部分析。第二个问题是区域分解方法中传输条件的优化问题。主要研究人员打算利用非线性偏微分方程的吸收边界条件来优化传输条件。这个项目将特别包含数值计算。第三个领域的工作是处理紧流形上非线性偏微分方程的长期存在性。目的是改进非线性偏微分方程解的局部存在性理论所给出的存在时间估计。数学技术将涉及先进的分析工具。本课题主要研究三个问题，两个问题与数值分析有关，一个问题与非线性偏微分方程的定性性质有关。建议的工作涉及非常有趣的数学问题，同时对应用也很重要。第一个和第二个问题处理的是庞大系统的数值计算，因此在许多领域都有应用，如环境科学（海洋学和气象学）以及医学（模拟人体血管系统中的血液流动）。虽然现在评价第三个问题的广泛社会影响还为时过早，但在过去的几十年里，偏微分方程的定性性质研究在许多领域（例如环境科学和人口动态，仅举几例）都有惊人的应用。这三个问题将巩固现代数学研究领域之间的重要联系，如数值分析、非线性偏微分方程和高级微局部分析。首席研究员将在应用数学家和纯数学家中传播项目结果。

Award Abstract 0504720, J Szeftel, Princeton UniversityTitle: Numerical computation and qualitative properties of nonlinear Partial Differential Equations The principal investigator proposes to work on three problems in applied mathematics. The first problem is related to the design of absorbing boundary conditions for nonlinear partial differential equations. The main goal is to obtain absorbing boundary conditions for systems such as those of fluid dynamics. This work will contain numerical computations as well as advanced microlocal analysis. The second problem deals with the optimization of the transmission conditions in domain decomposition methods. The principal investigator intends to use the absorbing boundary conditions designed for nonlinear partial differential equations to optimize the transmission conditions. This project will contain in particular numerical computations. The third area of work deals with long-time existence for nonlinear partial differential equations on compact manifolds. The aim is to improve the estimate on the time of existence given by the local existence theory for solutions of nonlinear partial differential equations. The mathematical techniques will involve advanced tools in analysis. This project focuses on three problems, two problems being related to numerical analysis and one to qualitative properties of nonlinear partial differential equations. The suggested works involve very interesting mathematical questions and at the same time are important for applications. The first and the second problem tackle with the numerical computation of huge systems and have therefore applications in numerous areas such as environmental sciences (oceanography and meteorology) as well as medicine (simulation of blood flows in human vascular system). While it is too soon to appraise the wide societal impact of the third problem, the last decades have seen stunning applications in many fields (e. g. environmental sciences and dynamic of populations to name just a few) of the study of qualitative properties of partial differential equations. These three proposed problems will solidify important connections between area of modern mathematical research such as numerical analysis, nonlinear partial differential equations and advanced microlocal analysis. The principal investigator will disseminate results of the project among both applied and pure mathematicians.