Small Scale and Singularity Formation in Fluids

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The project concerns the mathematical analysis of fluid mechanics and mathematical biology. Fluids are all around us, and better understanding of fluid motion is of importance in science and engineering. The project aims to advance understanding of small-scale formation in fluid motion, a process that appears in a wide range of applications and is related to development of turbulence. The project will also focus on analysis of chemotaxis, directed motion of cells or other biological agents in response to external chemical stimuli. Here the main goal is to understand and quantify how chemotaxis helps facilitate many biological processes from reproduction to immune system function. Many chemotactic processes take place in fluid, and interaction between the fundamental effects of diffusion, fluid flow and chemotaxis will also be studied. These problems are at the forefront of modern applied analysis and will require development of novel techniques that should be applicable in other settings. The project will also involve the training of junior researchers at a postdoctoral and graduate level.The first direction of the project is concerned with small scale formation and loss of regularity in patch solutions to the surface quasi-geostrophic (SQG) equation. The SQG equation appears in atmospheric science, where it is used to model large scale weather phenomena like temperature fronts. The PI and collaborators have recently discovered an intriguing structure in the evolution equation for curvature of the patch boundary, and plan to use this insight to obtain new results on ill-posedness and possible singularity formation in the bulk of the fluid. The second direction addresses small scale creation in solutions to the 2D Boussinesq system and 3D Euler equation. This direction also seeks to develop models suitable to gain insight into possible singularity formation in solutions of the 3D Euler and Navier-Stokes equations in the bulk, suggested by recent numerical simulations of Tom Hou. The third direction focuses on the coupled Keller-Segel-fluid system and explores the potential singularity suppression by fluid advection. The project aims to establish first rigorous results of this sort in the situation where advection is not passive and is not in a perturbative regime. 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 include novel comparison principles, methods of Fourier and functional analysis, asymptotic analysis techniques, as well as PDE 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.