# Displaying items by tag: NSF

## NSF CAREER Grant: Nonlocal Partial Differential Equations in Collective Dynamics and Fluid Flow

I am honored to be awarded an NSF CAREER grant (DMS #2238219) on a five-year project: **Nonlocal Partial Differential Equations in Collective Dynamics and Fluid Flow**.

**Abstract**

Collective behaviors are ubiquitous in nature and society. The mathematical study of collective dynamics has been active and fast-growing in recent decades. Many models have been proposed and analyzed to explain the intrinsic nonlocal interactions and the resulting complex emergent phenomena. These models are described by nonlocal partial differential equations. They have deep connections to classical systems in fluid dynamics. The goal of this project is to develop novel and robust analytical techniques to understand the collective behaviors driven by nonlocal structures. The training and professional development of graduate students and young researchers is an integral part of the project.

The project studies three families of partial differential equations with shared nonlocal structures that can affect the solutions of the equations: existence, uniqueness, regularity, and long-time asymptotic behaviors. The first problem is on the compressible Euler system with nonlinear velocity alignment, which describes the remarkable flocking phenomenon in animal swarms. Global phenomena and asymptotic behaviors of the system will be investigated, with a focus on the nonlinearity in the velocity alignment. The second problem is on the pressureless Euler system, aiming at the long-standing question of the uniqueness of weak solutions. The plan is to approximate the system by the relatively well-studied Euler-alignment system in collective dynamics. The third problem is on the Euler-Monge-Ampère system which is closely related to the incompressible Euler equations in fluid dynamics. The embedded nonlocal geometric structure of the system will be explored, with interesting applications in optimal transport and mean-field games.

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.

NSF award page on the grant DMS #2238219 |

## NSF Grant: Nonlocal Transport Equations in Fluids, Swarming, and Traffic Flows

I have been awarded an NSF grant (DMS #2108264) on a three-year project: **Nonlocal Transport Equations in Fluids, Swarming, and Traffic Flows**.

**Abstract**

Nonlocal models are relevant to many real-world phenomena and have been an area of active and growing research in recent decades. The development of a mathematical theory of nonlocal interactions plays a significant role in the understanding of complex structures, with rich applications in physics, biology, and social sciences. One example of the effects of nonlocal behavior found in nature is the collective dynamics in animal swarms, where small-scale interactions emerge into intriguing global phenomena. This project develops novel and robust analytical techniques for models that share similar nonlocality. These tools help to advance the understanding of the hidden structures of the models, and ultimately have an impact in applications, such as in traffic flow, where they can be used to study how to integrate nonlocal communications into a smart traffic network to improve efficiency and avoid traffic congestions. The training and professional development of graduate students is an integral part of the project.

The project studies three families of nonlocal transport equations. The first family includes the Euler-alignment system describing the flocking phenomenon for animal swarms. The goal is to establish a global well-posedness theory for the system in multi-dimensions, starting from imposing radial symmetry, and to apply the methodology to other models, such as the Euler-Poisson equations and more. The second includes a nonlocal transport equation which describes the evolution of the distribution of polynomial roots under repeated differentiation, the aim is to find a rigorous connection between this equation and the differentiation process. The last is a family of nonlocal traffic flow models, which have received extensive attention in the last decade, and are analyzed to understand the impact of the nonlocal interactions and how the nonlocal phenomenon can help to prevent traffic congestions.

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.

NSF award page on the grant DMS #2108264 |

## NSF Grant: Regularity and Singularity Formation in Swarming and Related Fluid Models

I have been awarded an NSF grant (DMS #1815667) on a three-year project: **Regularity and Singularity Formation in Swarming and Related Fluid Models**. It is translated to DMS #1853001 when I move to University of South Carolina.

**Abstract**

Swarming is a commonly observed complex biological and sociological phenomenon. The internal interaction mechanism attracts a lot of attention in physics, engineering, biology, and social sciences. This project is devoted to developing a unified mathematical theory towards the understanding of the swarming dynamics, as well as other nonlocal models that share similar structures. These models are widely considered in fluid mechanics, meteorology, astrophysics, biology, and ecology. The study of the regularity and singularity formations of these equations will provide a firm theoretical foundation for these applications, and also help consolidate the validity of these models in describing the natural phenomena.

The research will focus on understanding the nonlinear and nonlocal phenomena in swarming dynamics, and models having related structures in fluid mechanic. Three different but related models will be investigated. The first model is the Euler-Alignment system, which describes the flocking behavior in animal swarms. The goal is to develop a robust toolbox to analyze the nonlocal alignment operator and its balance with the drift nonlinearity. Similar behaviors are also observed in other fluid equations including porous medium flow, and surface quasi-geostrophic equations, which will be investigated using the same analytical techniques. The second model is the 2D inviscid Boussinesq equations. The global regularity is one of the outstanding problems in fluid dynamics. The idea is to construct solutions to capture the possible singularity formation, starting from some modified versions of the equations. The third model is the kinetic swarming system. The aim is to investigate the important relation between the kinetic equation and a variety of hydrodynamic limits. In particular, different alignment operators will be considered at the kinetic level. They are expected to lead to different macroscopic limits. All these three sub-projects will advance the mathematical understanding of nonlocal PDEs and related applications. They will also provide education and training to graduate and undergraduate students in this active 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.

NSF award page on the grant DMS #1853001 |