CAREER: Steady states and stability of nonlinear PDEs in fluids and interacting particle systems

The mathematical study of fluid dynamics and interacting particle systems plays a crucial role in science and engineering, where one of the main approaches is the analysis of the associated partial differential equations (PDEs). These equations are pertinent to a wide array of real-world applications and have been extensively researched. For instance, they are essential in understanding related phenomena in physics and biology, such as formation of severe weather conditions and collective animal behavior.

Additionally, they play a vital role in addressing some of the nation’s most pressing issues such as weather prediction, aircraft design and collective dynamics modeling. Developing mathematical theories in this area is both challenging and immensely valuable for comprehending intricate behaviors. The primary focus of this project is to investigate the steady states of these equations.

The main goal is to develop a comprehensive analytical framework that enhances the understanding of these steady states, including their existence, singular behavior, and stability. An integral part of this project is the educational component, which will offer training opportunities for students and provide platforms for professional development of young researchers in PDEs and analysis.

This project contains three interrelated directions. The first direction focuses on the three-dimensional (3D) incompressible Navier-Stokes equations (NSE), progressing toward a comprehensive understanding of an important class of steady states with scaling-invariance property. The plan is to first address the existence of these solutions and then investigate their singularity behavior and classification.

The second direction aims to understand the stability of steady states of the 3D incompressible NSE, building on the results from the first direction, where new analytical tools with more general applications are developed. The third direction studies the aggregation-diffusion equation. The goal is to study the steady states of the equation and whether the dynamics converge to the steady states, as well as the stability of the related interaction energy.

This project is jointly funded by the Applied Mathematics program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).

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.