CAREER: Transport Equations in Fluids and Biology: Singularity, Dynamics, and Mixing

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This project is devoted to a mathematical study of transport phenomena that are ubiquitous in nature. They relate to a transport of some substance, for example a pollutant or particulate material, by a flow, be it a fluid or gas flow or an intra-cellular flow. Mathematically, a certain scalar or vector field (e.g. density, vorticity, temperature) is carried by some velocity field. These phenomena can be described by partial differential equations (PDE), which often involve both nonlocal (dependence on the history of the flow or on the not necessarily close-by events) and nonlinear terms. Examples include the 2D Boussinesq equation that models large scale atmospheric and oceanic flows, and the diffusion-aggregation equation that models collective animal behavior. Due to the nonlocal and nonlinear nature of these equations, it is often unknown whether solutions exist globally in time or develop a finite-time singularity. Even in the cases where solutions are known to be global, their long-time behavior remains unclear for many equations. This project aims to develop novel analytical tools for a range of transport equations arising in fluid dynamics and biology, focusing on singularity, asymptotic, and mixing properties of the solutions. An integral part of the project is the educational component including developing advanced courses, supervising undergraduate research, and conducting a young researchers' workshop on nonlinear PDE. The workshop features mini-courses by established researchers and short talks by junior participants, aiming to introduce young researchers to the forefront of PDE research and facilitate collaborations. This project will advance the mathematical understanding of nonlocal PDE and their applications in fluids and biology. The project will also provide opportunities for education and training of junior researchers in this vibrant field.This project contains three different but related directions. The first direction is to obtain finite-time singularity formation for some fluid equations. The plan is to start with some one-dimensional model equations and prove finite-time singularity by establishing some kind of global control up to the blow-up time. The study of these model equations may shed new light on the full dynamics of fluid equations in higher dimensions. A second direction is the development of new tools for understanding the long-time dynamics of aggregation-diffusion equations, where the gradient flow structure plays an important role. A third direction is mixing by incompressible flows, and the goal is to study how fast the density can get mixed given some quantitative constraint of the velocity field.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.