Several Fundamental Questions in Hyperbolic Balance Laws

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Hyperbolic balance laws arise in the modeling of the nonlinear motion of fluid flows. The extremely complicated structure of this class of systems is a source of many open questions with practical implications in a broad class of scientific and engineering models. This research will focus on several fundamental questions, with the goal of bringing fresh insights and developing new tools to break ground in the field. The project aims to enhance understanding of these complicated but important phenomena. In addition to its scientific goals, this project aims to foster international collaborations and train graduate students and young researchers.The research project follows three main themes. The first is to look for sharp conditions for the global existence of smooth solutions to compressible, non-isentropic, Euler equations in one dimension with large initial data, and to continue the effort of deriving a sharp density lower bound estimate for generic large solutions. The second is to characterize the passage of singular limits in the isentropic approximation process for both inviscid and viscous fluids, and to derive error estimates in such approximations. The last is to establish the mathematically rigorous validity of nonlinear dynamical Rayleigh-Taylor type instability in compressible fluids subject to gravity, in particular for the compressible Navier-Stokes-Fourier system, where heat transfer plays an essential role. These research activities aim to break ground in several fields where theories are lacking, pave roads in areas that are outside current methodology, and advance understanding of the behavior of solutions in nonlinear hyperbolic balance laws.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.