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

该奖项与美国和英国的数学家团队合作，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的长期数学现象，例如激波的相互作用、涡流层的稳定性以及粘性消失时边界层的行为。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应用于涉及许多相互作用的多种物理、生物、金融和社会系统。该奖项将为学生提供参与合作研究的机会，并在美国和英国的不同机构举行研讨会。这一合作研究项目将开发创新的数学方法和技术，以研究跨尺度的非线性偏微分方程稳定性问题，包括双曲型守恒律中的渐近、量化和结构稳定性问题，动力学方程，以及流体-粒子(基于代理)模型中的相关多尺度应用。这项研究主要集中在以下四个相互关联的目标上：(1)反射/绕射激波模式的稳定性分析，重点是气体动力学中的激波反射-绕射问题；(2)涡旋片、接触不连续性和其他特征不连续性的稳定性分析；(3)颗粒到连续介质极限的稳定性分析，包括量化两两相互作用的渐近/平均场/大时间极限和多介质或多颗粒系统之间一般相互作用的颗粒极限；(4)渐近极限的稳定性分析，重点关注从多维可压缩粘性到无粘流的解在大初始数据下的零粘性极限。该项目将带来对这些基本科学问题的新理解和有益的相互促进，并朝着跨多尺度应用的非线性偏微分方程的非线性稳定性理论取得重大进展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.