From Quantum Many-body Dynamics to Fluid Equations and Back

The analysis, simulation, & applications of the nonlinear fluid equations like the Euler equations or the Navier-Stokes equations, is an important (if not a vital) part of many areas of Science, Technology, Engineering, and Mathematics (STEM). The research in this project concerns a variety of projects on the rigorous derivations of these macroscopic continuum equations from basic microscopic quantum particle models and elucidates how the macroscopic fluid-defining quantities like pressure or viscosity emerge from the averaging of microscopic quantities. Examples of the boson particles we study includes the nitrogen and oxygen molecules (99.03% volume of air) and 99.95% of the water molecules. The number of particles in these many-body systems is on the order of magnitude of the Avogadro constant, which make the microscopic simulation of such systems impossible. The mathematical justification of these macroscopic continuum limits for the many-body systems they are supposed to describe, is therefore an issue of fundamental scientific importance. The principal investigator is committed to introducing undergraduate and graduate students to experiments and cutting-edge mathematics, advising PhD students and mentoring postdoctoral researchers. The particular scope of this research project is to investigate several problems concerning the fine properties of solutions to the time-dependent many-body Schrödinger equation when the particle number tends to infinity and the Planck constant tends to zero. This research project encompasses three broad directions. The first direction concerns the proof of the classical incompressible Euler equations as a direct limit of quantum many-body dynamics and find the microscopic quantity corresponding to the macroscopic Mach number. The second direction is to rigorously extract the hierarchy structure for the compressible Euler equations induced by quantum many-body dynamics and identify the microscopic quantity which becomes the macroscopic Knudsen number. The third direction turns to the study of the optimal well/ill-posedness separation and the fine nonlinear structure of solutions regarding the important mesoscopic Boltzmann equations via new dispersive methods. The PI and collaborators use techniques from harmonic analysis, probability, and spectral theory to analyze these problems. 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.