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

Nonlinear partial differential equations (NPDEs) are at the heart of many scientific advances, with both length scales ranging from sub-atomic to astronomical and timescales ranging from picoseconds to millennia. Stability analysis is crucial in all aspects of NPDEs and their applications in Science and Engineering, but has grand challenges.

For instance, when a planar shock hits a wedge head on, a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of shock reflection-diffraction configurations was reported by Ernst Mach in 1878, and later experimental, computational, and asymptotic analysis has shown that various patterns of reflected-diffracted shocks may occur. Most fundamental issues for shock reflection-diffraction have not been understood. The global existence and stability of shock reflection-diffraction solutions in the framework of the compressible Euler system and the potential flow equation, widely used in Aerodynamics, will be a definite mathematical answer.

Another example arises in the analysis of mean field limits, a powerful tool in applied analysis introduced to bridge microscopic and macroscopic descriptions of many body systems. They typically involve a huge number of individuals (particles), such as gas molecules in the upper atmosphere, from which we want to extract macroscopic information. Multi-agent systems have become more popular than ever. In addition to their new classical applications in Physics, they are widely used in Biology, Economy, Finance, and even Social Sciences. One key question is how this complexity is reduced by quantifying the stability of the mean field limit and/or their hydrodynamic approximations.

By forming a distinctive joint force of the UK/US expertise, the proposed research is to tackle the most difficult and longstanding stability problems for NPDEs across the scales, including asymptotic, quantifying, and structural stability problems in hyperbolic systems of conservation laws, kinetic equations, and related multiscale applications in transonic/viscous-inviscid/fluid-particle models. Through this rare combination of skills and methodology across the Atlantic, the project focuses on four interrelated objectives, each connected either with challenging open problems or with newly emerging fundamental problems involving stability/instability:

Objective 1. Stability analysis of shock wave patterns of reflections/diffraction with focus on the shock reflection-diffraction problem in gas dynamics, one of the most fundamental multi-dimensional (M-D) shock wave problems;

Objective 2. Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities for M-D hyperbolic systems of conservation laws, especially including the equations of M-D nonisentropic thermoelasticity in the Eulerian coordinates, governing the evolution of thermoelastic nonconductors of heat;

Objective 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 systems;

Objective 4. Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from M-D compressible viscous to inviscid flows with large initial data.

These objectives are demanding, since the problems involved are of mixed-type and multiscale, as well as M-D, nonlocal, and less regular, making the mathematical analysis a formidable task. While many of the problems in the project have been known for some time, it is only recently that their solutions seem to have come within reach; in fact, part of the project would have been inconceivable prior to 2010. The simultaneous study of problems associated with the four objectives above will lead to a more systematic stability analysis for NPDEs across multiscale applications.