CAREER: Regularity estimates for elliptic and parabolic equations

这个数学研究项目是在偏微分方程的一般领域。重点研究椭圆型和抛物型偏微分方程的正则性，以及它们与这类问题的适定性之间的联系；该项目的部分内容涉及非局部扩散。本课题研究的偏微分方程来源于处理不连续过程的随机模型。金融学、物理学、种群动力学、化学和生物学中的许多模型都涉及非局部扩散。另一个相关的研究方向是局部或非局部扩散与平流之间的相互作用。物理学中有许多平流扩散模型目前在数学上还没有得到很好的理解。这种现象的典型例子出现在流体动力学模型方程的研究中。首席研究员Luis Silvestre将研究平流扩散方程的估计，旨在提高对流体动力学模型中产生的主动标量方程的理解。Silvestre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距离相互作用的物理模型中。它们也自然地出现在控制任何概率模型的方程中，这些模型的值可能会有很大的跳跃。一个常见的例子是金融数学中的股票价格建模，它可能偶尔发生突然变化。非局部方程出现在物理学、金融学、社会科学和生物学的无数模型中。它们的应用范围很广，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。近年来，非局部偏微分方程一般理论的发展取得了很大的进展，它的应用也越来越广泛。描述流体动力学的方程式提出了非常困难的数学问题。Luis Silvestre（该项目的首席研究员）还将研究运动流体中的局部和非局部扩散方程。对于该提案的教育部分，Silvestre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，并指导研究生和博士后。

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.