Generic Singularities and Fine Regularity Structure for Nonlinear Partial
PI: Dr. Tien Khai Nguyen (Associate Professor of Mathematics)
Support: National Science Foundation (NSF)
Period of Performance: September 1st, 2022 – August 31st, 2025
Budget: $164,000
Summary: The present project proposes a series of fundamental questions together with steps to their resolutions, necessary for advancing and developing the regularity theory and generic singularity for weak solutions to nonlinear and nonlocal PDEs. The main parts of the proposal can be summarized as follows:
(i) Develop a quantitative analysis of the number of shocks, provide a detailed description of shock formation and wave breaking for entropy weak solutions as well as trace their impact on generic regularity and stability results for various models of nonlinear waves and for hyperbolic systems of conservation laws.
(ii) Deepen the analysis of the metric entropy and study the SBV regularity of viscosity solutions to Hamilton-Jacobi equations, developing new techniques that will cover a wider class of equations.
(iii) Investigate the fine properties of generalized monotone functions, study the propagation of singularities and establish both new regularity estimates and controllability results for Hamilton-Jacobi equations.