CAREER: Front Propagations and Viscosity Solutions

职业：前沿传播和粘度解决方案

• 批准号: 1843320
• 负责人: Hung Tran
• 金额: $ 42.5万
• 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: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1843320&HistoricalAwards=false
• 关键词: CAREER; Front; Propagations; Viscosity; Solutions

This project concerns some nonlinear partial differential equations (PDEs) that appear naturally in physics, economics, and engineering and that arise, for example, in the study of crystal growth, composite materials, combustion, game theory, and optimal control theory. The equations studied have deep connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, and inverse problems. The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the PDEs under investigation. A key object of the research is a crystal growth model, in which the crystal grows in both horizontal direction by adatoms, and in vertical direction by nucleation in a supersaturated media. To make practical use of the model, it is extremely important to understand deeply the qualitative and quantitative aspects of the growth speed of the crystal. An integral part of the project is the educational component including bringing up the number of graduate PDE students at University of Wisconsin-Madison through various activities. The incoming graduate students interested in PDE are encouraged to participate in the principal investigator's PDE reading seminar, and to interact more with their peers and postdocs in the area. In term of undergraduate training, the principal investigator plans to increase the interest of University of Wisconsin-Madison undergraduate mathematics majors in the study of Analysis and PDE through some individual mentoring plans, and two undergraduate summer schools. Besides, the principal investigator plans to organize two conferences in nonlinear PDE and related topics for early career researchers.The proposed research involves two themes. The first is about a level-set mean curvature equation with driving and source terms, and applications to a crystal growth model, in which each level set of the unknown evolves in time by its mean curvature with unit constant force. The second involves Eikonal equations, and homogenization, where the zero-level set of the unknown moves in a periodic environment with highly oscillatory normal velocity. The principal investigator and his collaborators have recently developed some new approaches, which provided solutions to several open problems in these two themes and related areas. The new approaches are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the field of nonlinear PDE and viscosity solutions.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.

该项目涉及某些非线性偏微分方程（PDE），这些方程在物理，经济学和工程中自然出现，例如在研究晶体生长，复合材料，燃烧，游戏理论和最佳控制理论的研究中。所研究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，几何，同质化理论和反问题。该项目的主要目标是发现新的基本原则和一般方法，以了解正在研究的PDE解决方案的特性。研究的关键对象是晶体生长模型，其中晶体通过Adatoms沿两个水平方向生长，在过饱和培养基中通过成核沿垂直方向生长。为了实际使用该模型，深入了解晶体生长速度的定性和定量方面非常重要。该项目不可或缺的一部分是教育部分，包括通过各种活动来培养威斯康星大学麦迪逊分校的PDE学生人数。鼓励对PDE感兴趣的即将到来的研究生参加首席调查员的PDE阅读研讨会，并与该地区的同龄人和博士后进行更多互动。在本科培训期间，首席研究人员计划通过一些个人指导计划和两所本科暑期学校来研究分析和PDE的威斯康星大学麦迪逊大学本科生数学专业的兴趣。此外，主要研究人员计划在非线性PDE和早期职业研究人员的相关主题中组织两个会议。拟议的研究涉及两个主题。第一个是大约具有驾驶和源项的水平平均曲率方程，以及应用于晶体生长模型的应用，其中未知的每个级别集以其平均曲率和单位恒力的平均曲率及时演变。第二个涉及艾科纳尔方程和均匀化，其中未知的零级集在具有高振荡性正常速度的周期性环境中移动。首席调查员及其合作者最近开发了一些新方法，这些方法为这两个主题和相关领域的几个开放问题提供了解决方案。预计将在该项目中进一步开发新方法，从而为非线性PDE和粘度解决方案带来新的观点和见解。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来获得支持的。

项目成果

State-Constraint Static Hamilton--Jacobi Equations in Nested Domains

状态约束静态哈密顿--嵌套域中的雅可比方程

• DOI: 10.1137/19m1292035
• 发表时间: 2020
• 期刊: SIAM journal on mathematical analysis
• 影响因子: 2
• 作者: Kim, Yeoneung;Tran, Hung Vinh;Tu, Son
• 通讯作者: Tu, Son

Coagulation‐Fragmentation Equations with Multiplicative Coagulation Kernel and Constant Fragmentation Kernel

具有乘法凝聚核和恒定碎片核的凝聚-碎片方程

• DOI: 10.1002/cpa.21979
• 发表时间: 2022
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Tran, Hung V.;Van, Truong‐Son
• 通讯作者: Van, Truong‐Son

Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method

• DOI: 10.1016/j.jcp.2021.110828
• 发表时间: 2021-04
• 期刊: J. Comput. Phys.
• 作者: M. Klibanov;L. Nguyen;H. Tran

Level-set forced mean curvature flow with the Neumann boundary condition

• DOI: 10.1016/j.matpur.2022.11.002
• 发表时间: 2021-08
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: Jiwoong Jang;Dohyun Kwon;Hiroyoshi Mitake;H. Tran

Effective Fronts of Polytope Shapes

多面体形状的有效前沿

• 发表时间: 2020
• 期刊: Minimax Theory and its Applications
• 影响因子: 0.7
• 作者: Jing Wenjia;Tran Hung V.;Yu Yifeng
• 通讯作者: Yu Yifeng