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 m