Turbulence, Shocks, and Stability in Fluids and Plasmas

This project will develop the mathematical theory of liquids, gases, and plasmas, which are ubiquitous in problems from engineering, meteorology, aerodynamics, and more. These fundamental states of matter exhibit a wide range of behaviors, including turbulent and chaotic behavior, shock waves, and stability. The first goal is to advance the mathematical theory of turbulent fluids, which may be observed everywhere from the wakes of vehicles to the atmosphere. Second, this project will study shock waves, or the apparent discontinuities in properties such as density and flow velocity, which are observed in astrophysical plasmas and more. Finally, the project will investigate the stabilizing properties of fluids and plasmas near background shears, which may be used in a number of important applications to control the behavior of the fluid or plasma. This project also includes training and mentoring opportunities for graduate students and the organization of conferences and seminars. The first portion of this project will construct dissipative solutions of the incompressible Euler and Navier-Stokes equations, as well as other models of fluid and plasmas. Intermittency, wavelet-based iterations, and more will play a key role in the analysis. The second portion of this project will begin by building small-amplitude kinetic shock solutions to the Boltzmann and Landau equations which approximate traveling wave solutions of the compressible Navier-Stokes equations. Tools from the study of compressible fluids, the hydrodynamic limit, and kinetic theory will be developed and then used to investigate models of dilute charged particles, such as the Vlasov-Maxwell-Boltzmann system. Finally, this project will study hydrodynamic and magnetohydrodynamic stability and control. Stabilizing mechanisms, mixing, and enhanced dissipation are often observed in the vicinity of shear flows and will be used in a novel way to solve control problems for fluids and plasmas. 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.