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，周期性是一种相当有限的结构），并且需要研究平均（介观和宏观）行为。对于复杂的现象，通常情况下，大多数可用的信息都是“随机的”（随机）而不是“精确的”（确定性）。此外，将几个物理量的波动导致方程与“奇异”（白色噪声类型）和“随机”的依赖于一些变量。在这种背景下，随机均匀化和随机偏微分方程成为自然的数学对象。从数学的角度来看，随机性与对状态变量的奇异依赖性和缺乏紧凑性有关，这两种情况都会引起具有挑战性的数学问题。克服这些问题需要发展新的方法和技术。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。需要新的工具和想法来进一步研究这些问题，并确定影响实验观察和理论分析行为的参数的所有相关制度/尺度。PI建议继续他的计划，以开发研究连续和统计物理，生物学，工程等中出现的非线性确定性和随机偏微分方程的方法。重点是理论的发展（i）时空随机介质中的非线性，抛物/椭圆和双曲偏微分方程的均匀化及其在平均场博弈和前沿传播中的应用;（ii）弱（随机粘性）完全非线性的解决方案，（退化）抛物型随机偏微分方程，以及（iii）数学生物学中适应动力学模型的分析。发展数学工具来研究多尺度环境中的复杂现象，特别是当唯一可用的信息/数据通常是统计（随机）时，是最重要的。 非线性，一阶和二阶，随机偏微分方程和随机均匀化出现在各种现象和应用的模型中，包括平均场游戏，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，部分观测的随机控制，金融数学等。随机粘性解的理论是一个重要的问题。它允许研究一类全新的完全非线性随机偏微分方程。随着这一主题的进一步发展，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，在应用领域发挥至关重要的作用。在随机介质中的均匀化研究又重新引起人们的兴趣。已经开发和拟议开发的新工具和方法有望成为该领域的标准方法。 在数学生物学中，所提出的工作有望提高对自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、重要的和非常活跃的。PI目前有四名研究生（两名女性）和两名博士后（一名女性），计划继续开展教育和培训活动，旨在培养研究拟议领域问题以及非线性偏微分方程的高素质研究人员。一般。