Stability Analysis for Nonlinear Partial Differential Equations Across Multiscale Applications
Supported by US NSF and UK EPSRC (>$1m).
PROJECT SUMMARY
This collaborative research project will develop innovative
mathematical methods and techniques to solve the outstanding stability
problems of 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 proposed research is 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, and will provide education and
training to students in the exciting research field of applied
mathematics.
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J. A. Carrillo, R. Shu, Existence of radial global smooth solutions to the pressureless Euler-Poisson equations with quadratic confinement, Preprint.
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