Project Grant 2406620

This $159,640 federal Project Grant awarded by the National Science Foundation (NSF) under the Mathematical and Physical Sciences (CFDA 47.049) program supports fundamental research at the University of Rochester. The research aims to derive macroscopic fluid equations like the Euler and Navier-Stokes equations from microscopic quantum many-body dynamics, with a focus on understanding how macroscopic quantities like pressure and viscosity emerge from microscopic particle behavior. The project encompasses three main directions: 1) proving the classical incompressible Euler equations as a limit of quantum many-body dynamics, 2) rigorously extracting the hierarchy structure for compressible Euler equations from quantum many-body systems, and 3) studying the optimal well/ill-posedness and nonlinear structure of Boltzmann equations using dispersive methods. The research employs techniques from harmonic analysis, probability, and spectral theory, and is expected to enhance fundamental scientific understanding with potential applications across science, technology, engineering, and mathematics. This award reflects NSF's mission and has been deemed worthy of support based on its intellectual merit and broader impacts.