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Research Project - EP/V051121/1
DMS(NSF)-EPSRC: Stability Analysis for
Nonlinear Partial Differential Equations across Multiscale Applications
Principal Investigator: Gui-Qiang G. Chen
Co-Investigator: Jose A. Carrillo
US Team:
Mikhail Feldman (Wisconsin-Madison), Pierre-Emmanuel Jabin (Penn
State), Alexis F. Vasseur (UT-Austin), and Dehua Wang (Pittsburgh)
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.
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