Evolution Equations and Elliptic Boundary Problems in PDE

这个数学研究项目将研究偏微分方程中的问题，包括演化方程和平稳问题。研究的演化方程包括非欧几里得空间上的非线性波动方程、非线性薛定谔方程、Euler和Navier-Stokes方程（NLS）。具体目标包括研究非线性波动方程的强色散估计，初始数据满足非纯粹Sobolev空间估计条件的NLS的局部和全局解，非线性波动和NLS方程的驻波和行波解及其稳定性，Navier-Stokes方程解的小粘度极限，以及粗糙边界区域上的Euler和Navier-Stokes方程。研究的平稳问题包括粗糙区域上的椭圆边界问题和弱有界几何流形的分析。具体的目标包括研究Fredholm理论对尽可能大的一类一致可整域的管理，在这种非光滑环境中寻找正则椭圆边界问题概念的适当的一般版本。这建立在Hofmann和Mitrea对弦弧域（也称为SKT域）类的研究基础上，并期待更一般的类，包括包含的域。对这些主题的研究将导致对偏微分方程有用的分析工具的发展。这些可能包括奇异积分算符、椭圆算符的调和分析、几何测度理论和奇异摄动理论等领域的新成果。这个偏微分方程领域的数学研究项目涉及一系列科学问题。Navier-Stokes方程与流体流动的研究有关，既包括流体如何流过管道，也包括流体如何输送溶剂，以及管壁光滑或粗糙的管道中流动的不同行为。非线性薛定谔方程模拟了各种现实生活中的现象，并应用于诸如光缆，浅水中的表面波以及被称为玻色-爱因斯坦凝聚物的低温物质的微妙状态等主题。粗糙域上方程研究的潜在应用包括对隐藏结构的研究，例如地球中物质性质的突然转变，通常跨越粗糙的未知边界，其位置的知识可能具有重大价值。拟议项目中发展的数学技术将有可能影响这些领域。在过去的四年里，泰勒已经指导了三名博士生，他们已经开始对这些问题做出自己的贡献，他现在正在培养一名新的学生。

This mathematics research project will investigate problems in partial differential equations, including both evolution equations and stationary problems. Evolution equations to be investigated include nonlinear wave equations on non-Euclidean spaces, nonlinear Schrodinger equations, and Euler and Navier-Stokes equations (NLS). Specific goals include investigation of strong dispersive estimates for nonlinear wave equations, local and global solutions to NLS whose initial data satisfy conditions not given purely in terms of Sobolev space estimates, standing wave and traveling wave solutions to nonlinear wave and NLS equations and their stability, small viscosity limits of solutions to Navier-Stokes equations, and Euler and Navier-Stokes equations on regions with rough boundary. Stationary problems to be investigated include elliptic boundary problems on rough domains and analysis on manifolds with weakly bounded geometry. Specific goals include investigation of Fredholm theory for as large a class of uniformly rectifiable domains as can be managed, looking for an appropriately general version of the notion of a regular elliptic boundary problem in this nonsmooth context. This builds on work with Hofmann and Mitrea on classes of chord-arc domains (also called SKT domains), and looks forward to more general classes, including domains with inclusions. Investigations of these topics will result in the development of useful analytical tools for partial differential equations. These will likely include new results in the areas of singular integral operators, harmonic analysis of elliptic operators, geometric measure theory, and singular perturbation theory.This mathematics research project in the area of partial differential equations makes contact with a range of scientific issues. The Navier-Stokes equations bear on the study of fluid flows, both how a fluid will flow through a pipe and how that flow will transport solvents, and what sorts of different behaviors govern flows in pipes with smooth or rough walls. Nonlinear Schrodinger equations model various real-life phenomena and have applications to subjects such as optical cables, surface waves in shallow water, and a delicate state of low temperature matter known as a Bose-Einstein condensate. Potential applications of the study of equations on rough domains include the investigation of hidden structures, such as sudden transitions in the nature of materials in the earth, often across rough, unknown boundaries, the knowledge of whose locations can be of substantial value. The mathematical techniques developed in the proposed project will have the potential to impact these areas. Taylor has directed three Ph.D. students in the past four years, who have begun to make their own contributions to such problems, and he is now beginning to train a new student.