Asymptotic properties of steady Navier-Stokes flows

The Navier-Stokes equations are used to model the motion of certain fluids and gases and are essential in weather forecasting, aerospace engineering, geophysics and many other disciplines. The nonlinear nature of these equations allows them to generate complicated dynamics, but it also makes predicting fluid behavior quite challenging. In this project, the investigator will explore the extent to which certain steady-state solutions such as tornado models describe the macro-scale behavior of larger classes of flows, affording an understanding and prediction of otherwise intractable fluid systems. In addition, the investigator will contribute to workforce development by mentoring students and will participate in outreach events to foster connections between the mathematical community and the public. The investigator will explore several projects related to this objective. The first considers the dynamic stability of vortex models which possess multiple cells, e.g., a zone of updraft surrounding a central downdraft. The second explores the existence and stability properties of singular stationary solutions in the presence of physically motivated boundaries. The third project will explore the ubiquity of viscous eddies in corner Navier-Stokes flows. The last will examine the Eulerian nature of super-critically singular solutions with the aim of finding conditions under which the Euler equations, an alternative to the Navier-Stokes equations which omits viscosity, determine the leading order structure of certain viscous flows. A connection will be drawn to predictability in forecasting. 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.