DMS-EPSRC Collaborative Research: Stability Analysis for Nonlinear Partial Differential Equations across Multiscale Applications

DMS-EPSRC 协作研究：跨多尺度应用的非线性偏微分方程的稳定性分析

• 批准号: 2219434
• 负责人: Alexis Vasseur
• 金额: $ 12.49万
• 依托单位: University of Texas at Austin
• 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: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2219434&HistoricalAwards=false
• 关键词: DMS; EPSRC; Collaborative; Research; Stability

This award partners a team of US and UK mathematicians to use their combined expertise for the purpose of studying several challenging and longstanding mathematical phenomena in fluid dynamics, for example, the interaction of shock waves, the stability of vortex sheets, and the behavior of boundary layers as the viscosity vanishes. The project will also investigate questions involving the behavior of particle systems as the number of particles becomes infinite; the understanding of the collective behavior of these systems has applications to a variety of physical, biological, financial, and social systems involving many interacting agents. The award will provide opportunities for students to be involved in collaborative research and workshops to take place at various institutions in the US and the UK. This collaborative research project will develop innovative mathematical methods and techniques to study outstanding stability questions for nonlinear partial differential equations across the scales, including asymptotic, quantifying, and structural stability problems in hyperbolic conservation laws, kinetic equations, and related multiscale applications in fluid-particle (agent based) models. The research is focused mainly on the following four interrelated objectives: (1) Stability analysis of shock wave patterns of reflections/diffraction with focus on the shock reflection-diffraction problem in gas dynamics; (2) Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities; (3) Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent or many-particle systems; (4) Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from multi-dimensional compressible viscous to inviscid flows with large initial data. The project will lead to both new understanding of these fundamental scientific issues and beneficial cross-fertilization with significant progress towards a nonlinear stability theory of nonlinear partial differential equations across multiscale applications.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.

该奖项与美国和英国数学家团队合作，利用他们的综合专业知识来研究流体动力学中几个具有挑战性和长期存在的数学现象，例如冲击波的相互作用、涡流片的稳定性以及边界行为当粘度消失时分层。该项目还将研究当粒子数量变得无限时涉及粒子系统行为的问题；对这些系统集体行为的理解可应用于涉及许多相互作用主体的各种物理、生物、金融和社会系统。该奖项将为学生提供参与在美国和英国各机构举办的合作研究和研讨会的机会。 该合作研究项目将开发创新的数学方法和技术，研究跨尺度的非线性偏微分方程的突出稳定性问题，包括双曲守恒定律、动力学方程以及流体粒子中相关多尺度应用中的渐近、量化和结构稳定性问题（基于代理的）模型。研究主要集中在以下四个相互关联的目标上：（1）反射/衍射冲击波模式的稳定性分析，重点关注气体动力学中的冲击反射-衍射问题； (2) 涡片、接触间​​断和其他特征间断的稳定性分析； (3) 粒子到连续体极限的稳定性分析，包括量化成对相互作用的渐近/平均场/大时间极限以及多智能体或多粒子系统之间一般相互作用的粒子极限； (4)渐进极限稳定性分析，重点关注具有大初始数据的多维可压缩粘性流到无粘性流解的消失粘度极限。该项目将带来对这些基本科学问题的新理解，并在跨多尺度应用的非线性偏微分方程的非线性稳定性理论方面取得重大进展，带来有益的交叉融合。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。