Stabilizing Phenomenon for Incompressible Fluids

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This project seeks to understand an important universal stabilizing phenomenon concerning several incompressible fluids. The magnetic field stabilizes and damps electrically conducting fluids, a phenomenon observed in many physical experiments and numerical simulations. The temperature tames and stabilizes buoyancy driven fluids and helps the formation of stable structures in turbulent thermal convection. These are just two outstanding examples of a universal stabilizing phenomenon. The goal of this project is to fully comprehend this remarkable phenomenon and establish these observations as mathematically rigorous stability results on the models governing the dynamics of these flows. Understanding the proposed stability problems will help gain insight into many astronomical and meteorological phenomena such as Northern lights, solar flares, and severe weather events. The PIs will integrate the proposed research with the training of graduate students. In addition, the PIs will host events to get more K-16 students interested in these mathematical topics.This project focuses on the stability and large-time dynamics of the magnetohydrodynamic (MHD) equations near a background magnetic field and on the Boussinesq equations near the hydrostatic equilibrium. Mathematically these are extremely difficult problems. The fluid velocity in these MHD and Boussinesq systems is governed by the Euler or the Euler-like equations. The corresponding vorticity gradients could potentially grow rather rapidly in time. This makes the proposed stability problems appear to be impossible. The goal here is to solve these difficult problems by creating new strategies and innovative approaches. In particular, the PIs will reverse some of the classical approaches to the global regularity and stability problems on the MHD and the Boussinesq equations and treat the bad terms such as the Lorentz force and the buoyancy force in these models as good terms and exploit the smoothing and stabilizing effect of the magnetic field and the temperature. In addition, extensive numerical simulations will be performed to complement the theoretical studies.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.