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.

该奖项伙伴是美国和英国数学家团队使用其组合专业知识来研究流体动力学中的几种具有挑战性和长期的数学现象的目的，例如，冲击波的相互作用，涡流表的稳定性以及边界层的行为随着粘度而消失。该项目还将研究涉及粒子系统行为的问题，因为粒子的数量变得无限。对这些系统的集体行为的理解在涉及许多相互作用的代理的各种物理，生物，财务和社会系统上都有应用。该奖项将为学生提供合作研究和研讨会的机会，并在美国和英国的各个机构举行。 该协作研究项目将开发创新的数学方法和技术，以研究整个尺度上非线性偏微分方程的杰出稳定性问题，包括双曲线保护法，动力学方程，动力学方程以及相关的多构型应用程序中的流体疗法（基于代理）模型中的渐近，量化和结构稳定性问题。该研究主要集中在以下四个相互关联的目标上：（1）反射/衍射的冲击波模式的稳定性分析，重点是气体动力学中的冲击反射偏差问题； （2）涡流板，联系不连续性和其他特征性不连续性的稳定性分析； （3）粒子到连续限制的稳定性分析，包括用于成对相互作用的渐近/平均场/大时限和多粒子系统之间一般相互作用的粒子极限； （4）对渐近极限的稳定性分析，重点是从多维可压缩粘性到具有大初始数据的无粘性流的溶液的消失粘度极限。该项目将导致对这些基本科学问题的新理解和有益的交叉施用，并在多尺度应用中朝着非线性局部差分方程式的非线性稳定理论进行了重大进展。该奖项反映了NSF的法定任务，并认为通过基金会的智力和更广泛的影响，通过评估来评估NSF的法定任务。