NSF Award
2307097
Fluid models are used to make predictions about critical real-world systems arising in diverse fields including but not limited to meteorology, climate science, mechanical engineering, and geophysics. Simulations based on fluid models can, for example, be used to make predictions about the strength of a tornado or the stresses on an aircraft wing passing through turbulent air. The possibility that a mathematical model does not capture the full range of possible real-world scenarios is concerning if the predictions do not account for extreme events. Mathematically, this may occur if the model is unstable. This can be evident through a butterfly effect, in which a seemingly negligible change in a parameter leads to a wildly different outcome. Even more concerning is the specter of non-uniqueness, in which the same set of parameters may generate different dynamics. Consequently, a simulation could accurately describe one real-world scenario but not account for another possibly catastrophic one. In this project, the investigator will develop a broad understanding of the possible severity of these unstable dynamics. Students will be involved in this project. The project will include outreach efforts to promote mathematics in secondary schools and community colleges. <br/> <br/>The investigator will address the issue of how rapidly solutions to partial differential equations can and possibly must separate, primarily in the context of the Navier-Stokes equations, a system which models viscous incompressible fluid flow. Recent results suggest non-uniqueness for this system in physical classes of solutions. To assess the severity of non-uniqueness, estimates will be developed for the difference of two solutions having the same initial data. If the difference grows slowly, then one solution can be approximated from the other. If it grows rapidly, then the solutions quickly become uncorrelated, which is concerning when making predictions. New approaches for these estimates will be developed, e.g., through higher-order local time-regularity results. The investigator will additionally explore connections to an experimentally supported view of predictability, in which small scale instabilities do not instantly ruin macroscopic forecasts. Similar questions about non-uniqueness and separation arise in other partial differential equations models, which can come from the world of fluids, as with the surface quasi-geostrophic equation, but are not limited to it, as with the semi-linear heat or complex Ginzburg-Landau equations. The results for the Navier-Stokes equations will be adapted to these models, providing valuable information about the evolution of non-uniqueness in these equations and shedding light on how robust they are across partial differential equations with significantly varying structures.<br/><br/>This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).<br/><br/>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.
