Numerical computation and qualitative properties of nonlinear Partial Differential Equations

Princeton University, Princeton NJ

Investigators

Abstract

Award Abstract 0504720, J Szeftel, Princeton University Title: 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.