CAREER: Regularity estimates for elliptic and parabolic equations

职业：椭圆方程和抛物线方程的正则性估计

• 批准号: 1254332
• 负责人: Luis Silvestre
• 金额: $ 55万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-03-01 至 2019-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1254332&HistoricalAwards=false
• 关键词: CAREER; Regularity; estimates; elliptic; parabolic

This mathematics research project is in the general area of partial differential equations. The focus is on the study of the regularity properties of elliptic and parabolic partial differential equations, and their connections with the well posedness of such problems; parts of the project concern non local diffusions. The partial differential equations investigated in this project arise from stochastic models dealing with discontinuous processes. A number of models from finance, physics, population dynamics, chemistry and biology involve non local diffusions. Another related direction of research concerns the interaction between local or non local diffusion with advection. There are a number of advection-diffusion models from physics that are not currently well understood mathematically. Typical examples of such phenomena arise in the study of equations that model fluids dynamics. The principal investigator Luis Silvestre will study estimates for advection-diffusion equations geared towards an improved understanding of the active scalar equations arising from models in fluid dynamics. Silvestre will also investigate a number of questions related to fully nonlinear elliptic partial differential equations that occur in the study of zero-sum stochastic games.This mathematics research projects studies the behavior of so-called non-local equations. Such equations arise in any physical model with long range interactions. They also arise naturally as the equations governing any probabilistic model whose values may take long jumps. A common example is the modeling of stock prices in financial mathematics, which could sporadically take sudden changes. Non local equations appear in myriad of models from physics, finance, social sciences and biology. Their applications may range from the valuation of financial options, to the effective computation of protein docking, which is useful in the design of medicinal drugs, and even in the modeling of the flight of birds. The development of a general theory of non local partial differential equations has seen great progress in recent years, which goes side to side with an increasing number of applications. The equations describing the dynamics of fluids present very difficult mathematical questions. Luis Silvestre (the principal investigator in this project) will also study certain diffusion equations, local and non local, in moving fluids. For the educational part of this proposal, Silvestre will organize summer schools, design an improved partial differential equations class for undergraduates, as well as teaching a graduate class; Silvestre will also organize conferences, and will mentor graduate students and postdoctoral fellows.

这个数学研究项目是在偏微分方程的一般领域。重点是研究椭圆和抛物型偏微分方程的正则性，以及它们与这些问题的适定性的联系;该项目的部分内容涉及非局部扩散。在这个项目中研究的偏微分方程产生于处理不连续过程的随机模型。许多来自金融、物理、种群动力学、化学和生物学的模型都涉及到非局部扩散。另一个相关的研究方向涉及本地或非本地扩散与平流之间的相互作用。有许多物理学中的对流扩散模型目前在数学上还没有得到很好的理解。这类现象的典型例子出现在对流体动力学模型方程的研究中。首席研究员路易斯·西尔维斯特将研究对流扩散方程的估计，以更好地理解流体动力学模型中产生的活动标量方程。西尔维斯特还将研究一些与零和随机博弈研究中出现的完全非线性椭圆偏微分方程相关的问题。这个数学研究项目研究所谓的非局部方程的行为。这样的方程出现在任何具有长程相互作用的物理模型中。它们也自然地出现在任何概率模型的方程中，这些模型的值可能会发生很大的跳跃。一个常见的例子是金融数学中的股票价格建模，它可能偶尔会发生突然的变化。非定域方程出现在物理学、金融学、社会科学和生物学的无数模型中。它们的应用范围可以从金融期权的估值，到蛋白质对接的有效计算，这在药物设计中很有用，甚至在鸟类飞行的建模中也很有用。非局部偏微分方程一般理论的发展近年来取得了很大的进展，其应用也越来越广泛。描述流体动力学的方程提出了非常困难的数学问题。Luis Silvestre（该项目的首席研究员）还将研究移动流体中的某些局部和非局部扩散方程。对于这项建议的教育部分，西尔维斯特将组织暑期学校，为本科生设计改进的偏微分方程课程，以及教授研究生课程;西尔维斯特还将组织会议，并将指导研究生和博士后研究员。