Singularities and Long Time Dynamics in Models of Fluids and Reactive Processes

This project is composed of two parts. The main objective is the study of mathematical models of fluid motions, and more specifically creation of small-scale structures and singularities in fluids. This is a question of great importance in mathematics as well as in physics and engineering, as it is related to fluid turbulence and also explores how well the theoretical models describe real world phenomena in extreme situations.

The project will focus on singularity creation for motions of fluids in porous media (e.g. underground aquifers), in atmospheric science models, as well as for dynamics of fluids near walls and other boundaries. The second objective of the project is the study of propagation of reactive processes (e.g. forest fires) through combustive media. While the dynamics of such a process may intricately depend on small scale variations in the environment, the goal of this part of the project is to demonstrate that in many situations averaging of these variations over large regions results in a more regular and predictable large scale and long-term behavior of the process.

This part will also involve the study of propagation of bacterial colonies through nutrient-rich environments, and the enhancement of its speed due to the phenomenon of bacterial chemotaxis. The proposal will provide opportunities for the involvement of students and junior researchers in the research projects.

The primary focus of this project is the study of singularities and singular solutions for several nonlinear partial differential equations (PDEs) that serve as models of incompressible fluid dynamics. This includes motion of fluids in porous media on domains with boundaries, such as aquifers sitting on top of impermeable rocky layers; atmospheric science models such as generalized surface quasi-geostrophic (gSQG) equations; as well as Euler equations, modeling motions of ideal fluids, on planar domains with irregular boundaries.

In some of these models the relevant local well-posedness theories have not been found yet, so their development will also be an integral part of the project. A secondary focus of the project is a better understanding of large-scale behavior of reactive processes spreading through heterogeneous media, specifically development of a homogenization theory for the nonlinear reaction-diffusion PDE that models such processes occurring in multi-dimensional random media.

The goal is to show that under fairly general hypotheses, large scale behavior of solutions to this model is governed by much simpler homogenized PDE that capture the effects of the random variations in the medium averaging out in the long term. In addition, effects of chemotaxis on the speed of propagation of bacterial colonies through nutrient-rich environments will also be studied.

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.