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

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.