Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用

• 批准号: 1266383
• 负责人: Panagiotis Souganidis
• 金额: $ 34.9万
• 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: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266383&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Applications

The modeling of multi-scale phenomena necessitates the use of random media (periodicity is a rather restrictive structure for many applications) and requires the study of averaged (mesoscopic and macroscopic) behaviors. For complex phenomena, it is also very often the case that most of the available information is ``statistical'' (random) and not ``exact'' (deterministic). Furthermore incorporating the fluctuations of several physical quantities leads to equationsmwith ``singular'' (white noise type) and ``random'' dependence on some of the variables. In this context, random homogenization and stochastic partial differential equation become the natural mathematical objects. From the mathematical point of view, the randomness is associated with singular dependence on the state variables and lack of compactness both giving rise to challenging mathematical problems. Overcoming them requires the development of new methods and techniques. In biology, experiments at the molecular scale as well as new theories have led to new sophisticated mathematical models. Novel tools and ideas are needed to study these problems further and to identify all the relevant regimes/scales of the parameters affecting the experimentally observed and theoretically conjectured behavior.The PI proposes to continue his program to develop methods to study nonlinear deterministic and stochastic partial differential equation arising in continuum and statistical physics, biology, engineering, etc.. The emphasis is on the development of theories for (i) the homogenization of nonlinear, parabolic/elliptic and hyperbolic partial differential equation in spatio-temporal random media and applications to mean field games and front propagation and (ii) weak (stochastic viscosity) solutions of fully nonlinear, (degenerate) parabolic stochastic partial differential equation, and (iii) the analysis of models for adaptive dynamics in mathematical biology.The development of mathematical tools to study complex phenomena in multi-scale environments, especially when very often the only available information/data are statistical (random), is of the out most importance. Nonlinear, first- and second-order, stochastic partial differential equation and stochastic homogenization arise in models for a wide variety of phenomena and applications including mean field games, turbulence, phase transitions and front propagation in random media, nucleations in physics, macroscopic limits of particle systems, stochastic control theory, stochastic control with partial observations, financial mathematics, etc.. The theory of stochastic viscosity solutions is important. It allows for the study of a completely new class of fully nonlinear stochastic partial differential equation. As the subject develops further, it is expected that it will play a crucial role in applied areas by providing the necessary tools to analyze previously intractable models. There has been a resurgence in interest in homogenization in random media. The novel tools and methods that have already been and are proposed to be developed are expected to become the standardmethodology in the field. In mathematical biology, the proposed work is expected to enhance the understanding of concrete phenomena in adaptive dynamics. All the proposed areas of work are current, important and very active. The PI, who currently has four graduate students (two female) and two postdocs (one female) plans to continue his educational and training activities aiming towards the development of high quality researchers working in problems in the proposed areas as well as nonlinear partial differential equation in general.

多尺度现象的建模需要使用随机介质(对于许多应用来说，周期性是一个相当受限的结构)，并且需要研究平均(中观和宏观)行为。对于复杂的现象，大多数可获得的信息也往往是“统计的”(随机的)，而不是“确切的”(确定性的)。此外，将几个物理量的涨落合并在一起会导致方程对某些变量具有“奇异”(白噪声类型)和“随机”依赖关系。在此背景下，随机齐次化和随机偏微分方程成为自然而然的数学对象。从数学的角度来看，随机性与对状态变量的奇异依赖和紧凑性的缺乏有关，这两者都带来了具有挑战性的数学问题。克服这些问题需要开发新的方法和技术。在生物学中，分子尺度的实验以及新的理论导致了新的复杂的数学模型。需要新的工具和思想来进一步研究这些问题，并确定影响实验观察和理论推测行为的所有相关参数的状态/尺度。PI建议继续他的计划，开发方法来研究连续介质和统计物理、生物、工程等领域中出现的非线性确定性和随机偏微分方程。重点是关于(I)时空随机介质中非线性、抛物型/椭圆型和双曲型偏微分方程的齐次化及其在平均场游戏和前沿传播中的应用，(Ii)完全非线性(退化)抛物型随机偏微分方程弱(随机粘性)解的理论的发展，以及(Iii)数学生物学中自适应动力学模型的分析。最重要的是发展数学工具来研究多尺度环境中的复杂现象，特别是当通常唯一可用的信息/数据是统计(随机)的时候。非线性、一阶和二阶随机偏微分方程和随机齐次化在许多现象和应用中出现，包括平均场博弈、湍流、随机介质中的相变和前沿传播、物理学中的成核、粒子系统的宏观极限、随机控制理论、带部分观测的随机控制、金融数学等。随机粘性解的理论是重要的。它允许研究一类全新的完全非线性随机偏微分方程式。随着该学科的进一步发展，预计它将通过提供必要的工具来分析以前难以处理的模型，从而在应用领域发挥关键作用。人们对随机媒体中的同质化重新产生了兴趣。已经开发和提议开发的新工具和方法有望成为该领域的标准方法。在数学生物学方面，这项拟议的工作有望加强对适应动力学中具体现象的理解。所有拟议的工作领域都是当前的、重要的和非常积极的。国际和平研究所目前有四名研究生(两名女性)和两名博士后(一名女性)，他计划继续开展教育和培训活动，旨在培养高素质的研究人员，研究拟议领域的问题以及一般的非线性偏微分方程。