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

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，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，涡旋片的稳定性，以及黏度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可以应用于涉及许多相互作用因子的各种物理、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进行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来研究跨尺度非线性偏微分方程的突出稳定性问题，包括双曲守恒定律、动力学方程中的渐近、量化和结构稳定性问题，以及流体-颗粒（基于代理）模型中的相关多尺度应用。研究主要集中在以下四个相互关联的目标上：(1)反射/衍射激波型的稳定性分析，重点研究气体动力学中的激波反射-衍射问题；(2)涡片、接触不连续面及其他特征不连续面的稳定性分析；(3)粒子对连续统极限的稳定性分析，包括多智能体或多粒子系统间两两相互作用的量化渐近/平均场/大时间极限和一般相互作用的粒子极限；(4)大初始数据下多维可压缩黏性流到无黏性流解的渐近极限稳定性分析，重点是黏性极限消失。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肥，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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.