Nonlinear Partial Differential Equations and Applications

无论是在现实世界环境中，还是在关键的技术挑战中，比如物理系统中的热波动、机器学习中的算法随机性以及气候动力学中的天气模式建模，波动都无处不在。同时，这种复杂的系统受到大量的影响，并依赖于各种各样的参数和相互作用。此外，对于许多复杂的现象，大多数可用的信息往往是“统计的”（随机的），而不是“精确的”（确定性的）。对随机性和复杂动力学行为的相互作用的系统理解旨在揭示普遍的性质，而不管手头具体系统的许多细节如何。它的发展依赖于推导和分析通用概念的尺度限制，不仅捕获它们的平均行为，而且捕获它们的波动。随机偏微分方程是研究和理解波动作用的自然数学对象。在科学和技术中出现的另一个非常当前和重要的问题是对同时涉及几个不同长度尺度的问题的理论和计算分析。例如，在宏观尺度上理解、建模和准确预测材料的行为需要考虑它们的微观结构。现代材料科学不再只在一个尺度上考虑材料而忽略更细的尺度，而是越来越明确地同时处理给定材料在许多不同尺度上的模型。均匀化和多尺度方法是研究这类问题的数学理论的两个方面，分别是理论和计算。事实上，在随机环境中考虑均匀化是必要的，因为对于真实材料的建模来说，周期性是相当有限的。生长模型是概率论和数学物理中用于研究具有普遍标度极限的随机偏微分方程的自然数学模型。平均场博弈是研究社会经济科学中典型问题的理想数学结构，由于个体主体的前瞻性行为，社会经济科学不同于物理环境。在这种情况下，智能体的目标是优化某些标准，连同她/他的动态，依赖于其他智能体及其行为。代理人会做出反应、预测和制定策略，而不是简单地立即做出反应。应用的例子包括宏观经济和现代冲突的建模。在这两种情况下，大量的代理在随机演变的环境中进行战略互动，所有代理都对部分共同和部分特殊的激励做出反应，并且都试图同时预测其他代理的动态决策。例如，电信行业的一些平均场博弈模型自然是建立在网络（图）上的。这就需要开发数学工具来研究图上的方程，并理解它们在节点间的行为。本项目为研究生提供研究训练机会。该项目是PI计划的延续，该计划旨在开发新的方法和技术，用于在自然科学和社会科学与工程中出现的非线性一阶和二阶确定性和随机偏微分方程（分别为PDEs和SPDEs）的定性和定量研究。重点是(i)具有乘法“粗糙”路径依赖的偏微分方程；（ii）随机介质均质化；（iii）奇异域上的适定性；（iv）野外游戏；（五）增长模型的收敛性。非线性一阶和二阶偏微分方程具有粗糙的，特别是随机的时间依赖性，在波动的研究中自然出现。路径解理论的进一步发展是重要的，因为它允许研究新的非线性spde类别，并有望通过提供分析以前难以处理的模型的工具在应用领域发挥关键作用。定性和定量研究随机均匀化问题需要发展新的论点和方法来解决从周期到平稳遍历介质的紧性损失。对比例增长模型行为的分析，产生了新的方程。它们的适定性要求对黏度解理论中的一些经典工具进行改进。现在人们已经很好地理解了平均场博弈论的一个非常重要的元素是所谓的主方程，它是一个无限维的偏微分方程，它将个体和集体行为都包含在一个方程中。尽管在研究主方程在特殊噪声和普通噪声存在时的光滑解的性质方面取得了相当大的进展，但在没有前者的情况下，通常不存在光滑解。在此背景下的另一个重要问题是弱解概念的发展，在相关的应用非单调设置。在图方程的研究中，两个最重要的问题是顶点间的适定性和正确耦合条件的确定。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Fluctuations are ubiquitous both in real world contexts and in key technological challenges like, among others, thermal fluctuations in physical systems, algorithmic stochasticity in machine learning and modeling of weather patterns in climate dynamics. At the same time, such complex systems are subject to an abundance of influences, and depend on a large variety of parameters and interactions. In addition, for many complex phenomena most of the available information is very often "statistical" (random) and not "exact" (deterministic). A systematic understanding of the interplay of stochasticity and complex dynamical behavior aims at unveiling universal properties, irrespectively of the many details of the concrete systems at hand. Its development relies on the derivation and analysis of universal concepts for their scaling limits, capturing not only their average behavior, but also their fluctuations. Stochastic partial differential equations are the natural mathematical object to study and understand the role of fluctuations. Another very current and important issue arising in science and technology is the analysis, both theoretically and computationally, of problems that involve several disparate length-scales at once. For example, understanding, modeling and accurately predicting the behavior of materials at the macroscopic scale necessitates to consider their microscopic structure. Instead of considering materials in one scale and neglecting the finer scales, modern materials science increasingly explicitly and concurrently deals with models of a given material at many different scales. Homogenization and multiscale approaches are the two, respectively theoretical and computational, facets of the mathematical theory to study such problems. As a matter of fact, it is necessary to consider homogenization in random environments since periodicity is rather restrictive for the modeling of real materials. Growth models are natural mathematical models used in probability and mathematical physics to study stochastic partial differential equations exhibiting a universal scaling limit. Mean-field games are the ideal mathematical structures to study the quintessential problems in the social-economic sciences, which differ from physical settings because of the forward looking behavior on the part of individual agents. In this context, an agent aims to optimize certain criteria which, together with her/his dynamics, depend on the other agents and their actions. Agents react, anticipate and strategize instead of simply reacting instantaneously. Examples of applications include the modeling of the macro-economy and conflicts in the modern era. In both cases, a large number of agents interact strategically in a stochastically evolving environment, all responding to partly common and partly idiosyncratic incentives, and all trying to simultaneously forecast the dynamic decisions of others. Some mean-field games models in, for example, telecommunications are naturally set on networks (graphs). This raises the need of the development of the mathematical tools to study equations on graphs and to understand their behavior across nodes. The project provides research training opportunities for graduate students.The project is a continuation of the PI's program to develop novel methodologies and techniques for the qualitative and quantitative study of nonlinear first- and second-order deterministic and stochastic partial differential equations (PDEs and SPDEs, respectively) arising in natural and social sciences and engineering. The emphasis is on (i) PDEs with multiplicative ``rough'' path dependence; (ii) homogenization in random media; (iii) well-posedness in domains with singularities; (iv) mean-field games; and (v) convergence of growth models. Nonlinear, first- and second-order partial differential equations with rough and, in particular, stochastic time dependence arise naturally in the study of fluctuations. The further development of the theory of pathwise solutions is important, for it allows to study of new classes of nonlinear SPDEs, and is expected to play a crucial role in applied areas by providing the tools to analyze previously intractable models. Studying qualitatively and quantitatively stochastic homogenization problems requires the development of novel arguments and methodologies to address the loss of compactness when going from periodic to stationary ergodic media. The analysis of the behavior of scaled growth models gives rise to new equations. Their well-posedness requires the refinement of some of the by now classical tools from the theory of viscosity solutions. It is by now well understood that a very important element of the mean-field game theory is the so-called master equation, an infinite-dimensional partial differential equation which subsumes in a single equation both the individual and collective behaviors of agents. In spite of considerable progress in the study of the properties of the smooth solution of the master equation in the presence of both idiosyncratic and common noises, less is known in the absence of the former in which case smooth solutions do not exist in general. Another important question in this context is the development of a notion of weak solution in the relevant for application nonmonotone setting. Two of the most important questions in the study of equations of graphs is the well-posedness across vertices and the identification of the correct coupling condition.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.