Stability Theory for Systems of Hyperbolic Conservation Laws

This project aims to develop mathematical tools for studying the stability theory of hyperbolic conservation laws, which model a wide range of physical systems, including traffic flow and fluid and gas dynamics. These systems often develop shocks or discontinuities, which pose significant challenges for their mathematical treatment. The primary objective is to investigate the uniform stability of viscous approximations of these models, particularly the Navier-Stokes equation, and design methods to mitigate the destabilizing effect of viscosity on shocks in fluid dynamics. The project will also offer training and mentorship opportunities for undergraduate and graduate students and postdoctoral researchers, to enhance their expertise in modeling, analysis, and communication. This project will further develop the theory for hyperbolic conservation laws, by extending the theory of weighted contraction with shifts, to obtain weak/BV principles, stability of BV solutions with respect to wild initial perturbations, and inviscid limit of physical viscous models as the Navier-Stokes equation. The project will build on recent developments in the theory of weighted contraction with shifts to solve a twenty-year-old conjecture in the case of isentropic flows. The project will also consider multi-D settings where the uniqueness of solutions is known to fail. While instabilities are expected due to turbulence, the lack of uniqueness seriously questions the prediction abilities of the models themselves. This pathology brings both opportunities and formidable challenges to the field. This research will develop a theory to reconcile instability and predictability for discontinuous flows at high Reynolds numbers. 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.