LEAPS-MPS: Analysis of Initial and Boundary Value Problems

LEAPS-MPS：初始值和边值问题的分析

• 批准号: 2247019
• 负责人: John Holmes
• 金额: $ 17.95万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-10-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247019&HistoricalAwards=false
• 关键词: LEAPS; MPS; Analysis; Initial; Boundary

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2)Differential equations have played a central role in the study of physical phenomena since Sir Isaac Newton described the motion of planetary objects using differential calculus, and are fundamental to the understanding of physics, engineering, biology, chemistry, and economics. Differential equations relate how quantities or systems change over space and time and are derived as models for real world situations. In applications such as optimal investing, infectious diseases, chemical reactions, and tsunamis, the differential models used are often of diffusive or dispersive type. The mathematical theory for diffusive and dispersive nonlinear partial differential equations is incomplete, even in basic setups such as when the region considered can be model as the half line or a bounded interval, which for example is the case when dealing with the ocean floor or a fiber optic cable. This project intends to further the mathematical study of these models, to aid scientists in their understanding of the underlining physical phenomena. The project will provide research and training opportunities for undergraduate and graduate students and for early career researchers, with a focus in promoting the participation of members of underrepresented groups in STEM. The central aim of this project is to study dispersive and diffusive equations on the half line and on bounded intervals using the Uniform Transform Method (UTM), which has recently been developed as an extension to the Fourier transform for initial and boundary data. The UTM allows to construct iterative maps via contour integrals in the complex plane but has not yet been successfully applied to the Nonlinear Schrödinger, Korteweg-de Vries, and Burgers equations, with initial and boundary value data in spaces of very low regularity. This project will use the UTM to extend to bounded intervals previous results obtained for unbounded domains in spaces of both very low and very high regularity, with the main goal of developing the techniques and theory necessary to study well-posedness and asymptotic behavior of the 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.

自2021年《美国救援计划法》（公法117-2）的全部或部分资助，自艾萨克·牛顿爵士（Isaac Newton爵士爵士）描述了使用差异计算的行星对象的运动，并且对物理学，工程，生物学，化学，化学和经济学的理解为基础。微分方程将数量或系统在空间和时间上发生变化，并作为现实世界情况的模型得出。在最佳投资，传染病，化学反应和海啸等应用中，使用的差异模型通常具有不同或分散类型。对于不同的和分散非线性偏微分方程的数学理论也不完整，即使在基本设置中，例如何时被认为是半线或有界间隔的模型，例如，在处理海底或光纤电缆时，这种情况就是这种情况。该项目旨在进一步对这些模型的数学研究，以帮助科学家理解强调物理现象。该项目将为本科生和研究生以及早期职业研究人员提供研究和培训机会，重点是促进代表性不足的STEM成员的参与。该项目的核心目的是使用统一变换方法（UTM）在半线和有限间隔上研究分散和扩散方程，最近已开发为初始和边界数据的傅立叶变换的扩展。 UTM允许在复杂平面中通过轮廓积分构建迭代图，但尚未成功应用于非线性schrödinger，korteweg-de vries和burgers方程，并在非常低的规律性的空间中具有初始和边界价值数据。 This project will use the UTM to extend to bounded intervals previous results obtained for unbounded domains in spaces of both very low and very high regularity, with the main goal of developing the techniques and theory necessary to study well-posedness and asymmetric behavior of the solutions.This award reflects NSF's statutory mission and has been deemed precious of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

项目成果

Nonuniform Dependence on the Initial Data for Solutions of Conservation Laws

守恒定律解对初始数据的非一致依赖

• DOI: 10.1007/s42967-023-00267-9
• 发表时间: 2023
• 期刊: Communications on Applied Mathematics and Computation
• 影响因子: 1.6
• 作者: Holmes, John M.;Keyfitz, Barbara Lee
• 通讯作者: Keyfitz, Barbara Lee