CAREER: Dynamical Aspects of Random Fluid Motion

A fluid is any substance that can flow and that takes the shape of its container. Unpredictable changes can affect the way fluids mix and move and this work seeks to elucidate these effects by developing new mathematical tools to study how random fluctuations affect fluids. The principal investigator will also explore how unpredictable changes can lead to chaotic and unstable behavior in fluids, which is important for understanding phenomena like turbulence. This effort will train future scientists to continue this work. Findings will be shared with other researchers and the public. By combining different areas of mathematics, a deeper understanding of how fluids behave will be achieved. This endeavor brings together scientists from different fields to share ideas and work together, potentially leading to new discoveries. This project aims to develop a rigorous mathematical framework for analyzing the influence of random fluctuations on mixing properties in fluid models, including advection-diffusion and nonlinear partial differential equations (PDEs). The research focuses on the interplay between deterministic and stochastic processes in shaping fluid behavior, examining two interconnected themes: (1) Microlocal approach to mixing by random velocities - by employing the new techniques using pseudodifferential operators, this project aims to develop a rigorous mathematical framework for analyzing the influence of random fluctuations on mixing properties in various fluid models, including advection-diffusion and nonlinear PDEs; (2) Lyapunov exponents in Infinite-Dimensional Systems - this project aims to advance techniques for studying Lyapunov exponents in infinite dimensional stochastic systems, particularly developing methods for proving the existence of projective stationary measures and the finiteness of Lyapunov exponents. Expected outcomes include an enhanced understanding of mixing and transport in fluid systems and a characterization of chaos in infinite-dimensional systems. The project also seeks to achieve a proof of exponential lower bounds for advection-diffusion equations in compact domains with quantitative dependence on diffusivity. The results of this research will advance the theoretical understanding of fluid dynamics and potentially generate practical implications for fields such as climate modeling, oceanography, and engineering. This project is jointly funded by the Applied Mathematics and the Analysis programs in the Division of Mathematical Sciences as well as the Established Program to Stimulate Competitive Research (EPSCoR). 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.