Several Fundamental Problems in Hyperbolic Balance Laws

Hyperbolic balance laws arise from the modeling of the nonlinear motion of fluid flows and are central to a wide range of natural and engineered systems – ranging from air conditioning efficiency to severe weather events like tornadoes and ocean tides. Despite their relevance, the complex mathematical structures of these systems continue to pose unresolved challenges. This project will address several fundamental problems in the theory of hyperbolic balance laws, aiming to develop new tools and advance our understanding of the dynamics of compressible fluids. In addition to its scientific goals, the project includes international collaboration and a strong education component focused on training graduate students and early-career researchers. There are three main themes in this research project. The first theme focuses on establishing sharp conditions for the global existence of smooth solutions to the one-dimensional compressible (non-isentropic) Euler equations with large initial data. This includes characterizing singularity developed from generic smooth data and continuing efforts to derive sharp lower bounds on density for generic large solutions. The second theme is to characterize the passage of singular limits in the isentropic approximation of both inviscid and viscous fluid models, with the goal of rigorously formulating the corresponding error estimates. The third theme aims to establish the mathematically rigorous validity of nonlinear dynamical Rayleigh-Taylor type instability in compressible fluids under the influence of gravity, especially for the compressible Navier-Stokes-Fourier system where heat transfer is essential. Collectively, these research efforts seek to advance the theoretical understanding of nonlinear hyperbolic balance laws, address longstanding open problems, and develop methodologies capable of tackling regimes beyond the reach of existing techniques. 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.