Manas Bhatnagar, Hailiang Liu and Changhui Tan

Abstract

This paper is concerned with the global wellposedness of the Euler-Poisson-alignment (EPA) system. This system arises from collective dynamics, and features two types of nonlocal interactions: the repulsive electric force and the alignment force. It is known that the repulsive electric force generates oscillatory solutions, which is difficult to be controlled by the nonlocal alignment force using conventional comparison principles. We construct invariant regions such that the solution trajectories cannot exit, and therefore obtain global wellposedness for subcritical initial data that lie in the invariant regions. Supercritical regions of initial data are also derived which leads to finite-time singularity formations. To handle the oscillation and the nonlocality, we introduce a new way to construct invariant regions piece by piece in the phase plane of a reformulation of the EPA system. Our result is extended to the case when the alignment force is weakly singular. The singularity leads to the loss of a priori bounds crucial in our analysis. With the help of improved estimates on the nonlocal quantities, we design non-trivial invariant regions that guarantee global wellposedness of the EPA system with weakly singular alignment interactions.

This work is supported by NSF grant DMS #1853001 and DMS #2108264

Trevor M. Leslie, and Changhui Tan

Abstract

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker-Smale dynamics. Our approach is largely inspired by the work of Brenier and Grenier [SIAM J. Numer. Anal, 35(6):2317-2328, 1998] on the pressureless Euler equations.

This work is supported by NSF grant DMS #1853001 and DMS #2108264

Eitan Tadmor, and Changhui Tan

Abstract

We study the global wellposedness of the Euler-Monge-Ampère (EMA) system. We obtain a sharp, explicit critical threshold in the space of initial configurations which guarantees the global regularity of EMA system with radially symmetric initial data. The result is obtained using two independent approaches -- one using spectral dynamics of Liu & Tadmor [Comm. Math. Physics 228(3):435-466, 2002] and another based on the geometric approach of Brenier & Loeper [Geom. Funct. Analysis 14(6):1182--1218, 2004]. The results are extended to 2D radial EMA with swirl.

This work is supported by NSF grant DMS #1853001 and DMS #2108264

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

Alexander Kiselev, and Changhui Tan

Abstract

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Stefan Steinerberger to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to described collective behavior of agents (such as birds, fish or robots) in mathematical biology. We consider periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data.

This work is supported by NSF grant DMS #1853001 and DMS #2108264