Viscosity Solutions: Beyond Well-Posedness Theory

粘度解决方案：超越适定理论

• 批准号: 1664424
• 负责人: Hung Tran
• 金额: $ 16.8万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664424&HistoricalAwards=false
• 关键词: Viscosity; Solutions; Beyond; Well; Posedness

This project concerns some nonlinear partial differential equations that appear naturally in physics, the social sciences, and engineering and that arise, for example, in the study of composite materials, combustion, game theory, traffic flow, and optimization. The equations considered have deep connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, inverse problems, and optimal control theory. The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the differential equations under investigation. One of the key objects of the research is homogenization theory, in which the models (say, of physical or social phenomena) are set in heterogeneous media and have many parameters varying on a small scale. If one zooms out and looks at the macroscopic scale, one often sees a simple effective (averaging) behavior. To make practical use of the models, it is extremely important to understand deeply the qualitative and quantitative aspects of this effective behavior.The project has two overarching themes: (i) going beyond mere well-posedness in order to understand fine properties of both the limiting process and the effective equation in homogenization theory (e.g., shape of the effective Hamiltonian, optimal rate of convergence), and (ii) advancing knowledge of the duality method and fully nonlinear elliptic equations (e.g., representation formulas of solutions, the vanishing discount problem, regularity). The principal investigator and his collaborators have recently developed new approaches in these (and related) topical areas, approaches that have already provided solutions to some open problems. These methods are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the field of viscosity solutions.

该项目涉及一些非线性偏微分方程，这些方程在物理，社会科学和工程中自然出现，例如在复合材料，燃烧，游戏理论，交通流量和优化的研究中出现。所考虑的方程式与许多其他数学领域有着深厚的联系，包括变化的计算，差异游戏，动力学系统，几何学，同质化理论，反问题和最佳控制理论。该项目的主要目标是发现新的基本原则和一般方法，以了解调查中微分方程的解决方案的特性。该研究的关键对象之一是均化理论，其中模型（例如，物理或社会现象）在异质媒体中设定，并且在小范围内具有许多参数。如果一个人放大并查看宏观量表，则通常会看到一种简单的有效（平均）行为。要实际使用这些模型，深入了解这种有效行为的定性和定量方面非常重要。该项目具有两个总体主题：（i）超越单纯的主题：为了了解限制过程的良好特性和均质理论的有效方程式（例如，有效的Hamiltonian vewool and Nonthine vernility vernine and contrace of Converline and contrime and contrimential and contrime and contrimential and contrace）的有效方程（椭圆方程（例如，解决方案的表示公式，消失的折扣问题，规律性）。首席调查员及其合作者最近在这些（及相关）主题领域开发了新方法，这些方法已经为某些开放问题提供了解决方案。预计这些方法将在该项目中进一步开发，从而为粘度解决方案领域带来新的观点和见解。

项目成果

Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model

临界凝固-破碎模型中 Hamilton-Jacobi 方程的大时间行为

• DOI: 10.4310/cms.2021.v19.n2.a8
• 发表时间: 2021
• 期刊: Communications in mathematical sciences
• 影响因子: 1
• 作者: Mitake, Hiroyoshi;Tran, Hung Vinh;Van, Truong-Son
• 通讯作者: Van, Truong-Son

A rigidity result for effective Hamiltonians with 3-mode periodic potentials

• DOI: 10.1016/j.aim.2018.06.017
• 发表时间: 2017-07
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: H. Tran;Yifeng Yu

A Note on Nonconvex Mean Field Games

关于非凸平均场博弈的注解

• 发表时间: 2018
• 期刊: Minimax theory and its applications
• 影响因子: 0.7
• 作者: Tran, Hung V.

Min–max formulas and other properties of certain classes of nonconvex effective Hamiltonians

某些类别的非凸有效哈密顿量的最小–最大公式和其他属性

• DOI: 10.1007/s00208-017-1601-8
• 发表时间: 2017
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Qian, Jianliang;Tran, Hung V.;Yu, Yifeng
• 通讯作者: Yu, Yifeng

On uniqueness sets of additive eigenvalue problems and applications

• DOI: 10.1090/proc/14152
• 发表时间: 2018-01
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Hiroyoshi Mitake;H. Tran