Small scale creation and singularities in fluids

This project addresses fundamental questions in fluid mechanics and chemotaxis modeling. Fluids are ubiquitous, with their finesse and complex behavior affecting both everyday life as well as more extreme phenomena such as tornados or hurricanes. This project aims to address a number of questions arising in the small-scale creation and singularity formation in fluids, which are linked with key processes such as mixing, energy dissipation, and turbulence. The project is also concerned with the interaction between fluid flow and chemotaxis, which is the motion of cells or other agents directed by the chemical gradient of a signaling compound. The project seeks to analyze the dispersing effect fluid advection can have in the context of chemotactic aggregation as well as study the effect of chemotaxis on biological reactions such as reproduction or workings of the immune system. Scaling laws will be derived to demonstrate how chemotaxis enables these processes, and questions on how their efficiency depends on a variety of parameters and the geometry of the environment will be studied. The training of graduate students and postdoctoral researchers will be an integral part of the project. The first direction of the project is concerned with small scale formation in solutions of the two-dimensional incompressible Euler equations. Emergence of large scale structures and small scale creation are two principal features of two-dimensional turbulence. The focus of the project is on the latter, specifically on robustness of mechanisms driving small scale creation and the influence of physical boundaries on this process. The second direction is focused on the incompressible porous media equation. This equation comes from geophysics, and models the flow of liquid such as oil or water through porous structures. The investigator plans to improve examples of solutions exhibiting derivative growth, as well as analyze a new scenario for singularity formation. The third direction focuses on coupled Keller-Segel-fluid systems and explores the potential singularity suppression by fluid advection. The goal is to better understand the ingredients involved in a newly discovered regularity mechanism. The final direction is concerned with the development of a new class of models addressing the effect of chemotaxis on biological reactions. The research is intended to go beyond regularity estimates and rigorously derive scaling laws that may be of interest in applications. A variety of techniques that will be deployed in all these directions include novel comparison principles, methods of Fourier and functional analysis, asymptotic analysis techniques, new functional inequalities, stochastic analysis, as well as regularity estimates. 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.