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

 Speaker: Ming Zhong (Illinois Institute of Technology)

Self organization (aka collective behaviors) occurs naturally in crystal formation, cell aggregation, social behaviors, etc. It is challenging and intriguing to understand self organization from the mathematical point of view. We offer a statistical/machine learning approach to explain self organization from observation data; moreover, our learning approach can aid in validating and improving the modeling of self organization. We develop a learning framework to derive physically meaningful dynamical systems to understand self organization from observation. We then investigate the steady state properties of our learned estimators. We also extend the learning approach for dynamical models constrained on Riemannian manifolds. We further improve our learning capability to infer interaction variables as well as interaction kernels. We study the effectiveness of our learning method on the NASA Jet Propulsion Laboratory's modern Ephemerides. Upon careful inspection of our model, we discover that it even captures potion of the general relativity effects. A complete learning theory on second-order systems is presented, as well as two new models on emergence of social hierarchy and concurrent emergence of flocking and synchronization.
 

Time: February 10, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Siming He and Changhui Tan

 Speaker: Roman Shvydkoy (University of Illinois Chicago)

Emergence is a phenomenon of formation of collective outcomes in systems where communications between agents has local range. In dynamics of swarms such outcomes often represent a globally aligned flock or congregation of aligned clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with non-integrable radial tails. Proving a similar statement for purely local interactions presents a major mathematical challenge. In this talk we will overview three programs of research directed on understanding the emergent phenomena: hydrodynamic topological interactions, kinetic approach based on hypocoercivity, and spectral energy method. We present a novel framework based on the concept of environmental averaging which allows us to obtain coercivity estimates leading to new flocking results.
 

Time: January 27, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Changhui Tan

 Speaker: Jing An (Duke University)

We will discuss the algebraic structure of a large class of reaction-diffusion equations and use it to study the long-time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. A new quantity named as the shape defect function is introduced to measure the difference between the profiles of the solution and the traveling waves. In particular, the positivity of the shape defect function, combined with a new weighted Hopf-Cole transform and a relative entropy approach, plays a key role in the stability arguments. The shape defect function also gives a new connection between reaction-diffusion equations and reaction conservation laws at the pulled-pushed transition. This is joint work with Chris Henderson and Lenya Ryzhik. 
 

Time: November 18, 2022 2:30pm-3:30pm
Location: LeConte 205
Host: Siming He

 Speaker: Federico Pasqualotto (Duke University)

Magnetic relaxation is a conjectured general procedure to obtain steady solutions to the incompressible Euler equations by means of a long-time limit of an MHD system. In some regimes, the magnetic field is conjectured to “relax” to a steady state of the 3D Euler equations as time goes to infinity.

In this talk, I will first review the classical problem of magnetic relaxation, connecting it to questions arising in topological hydrodynamics. I will then present a general construction of steady states of the incompressible 3D Euler equations by a long-time limit of a regularized MHD system. We consider the so-called Voigt regularization, and our procedure yields non-trivial equilibria on the flat 3D torus and on general bounded domains.

This is joint work with Peter Constantin. 
 

Time: October 28, 2022 2:30pm-3:30pm
Location: LeConte 205
Host: Siming He

McKenzie Black and Changhui Tan

Journal of Differential Equations, Volume 380, pp. 198-227 (2024)

Abstract

We consider the pressureless compressible Euler system with a family of nonlinear velocity alignment. The system is a nonlinear extension of the Euler-alignment system in collective dynamics. We show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are investigated, resulting in a variety of different asymptotic behaviors.

	doi:10.1016/j.jde.2023.10.044
	Download the Published Version
	This work is supported by NSF grants DMS #2108264 and DMS 2238219
	This work is supported by a UofSC VPR SPARC grant.

 Speaker: Pierre-Emmanuel Jabin (Pennsylvania State University)

We introduce a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension 2 together with a partial result in dimension 3. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted L p norms of the marginals or observables of the system uniformly in the number of particles. This allows to treat very singular interaction kernels between the particles, including repulsive Poisson interactions. This is a joint work with D. Bresch and J. Soler. 
 

Time: December 2, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

 Speaker: Amir Sagiv (Columbia University)

In many scientific areas, deterministic models (e.g., differential equations) use numerical parameters. Often, such parameters might be uncertain or noisy. A more honest model should therefore provide a statistical description of the quantity of interest. Underlying this numerical analysis problem is a fundamental question - if two "similar" functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? We will first show how the probability density function (PDF) of the quantity of interest can be approximated. We will then discuss how, through the lense of the Wasserstein-distance, our problem yields a simpler and more robust theoretical framework.

Finally, we will take a steep turn to a seemingly unrelated topic: the computational sampling problem. In particular, we will discuss the emerging class of sampling-by-transport algorithms, which to-date lacks rigorous theoretical guarantees. As it turns out, the mathematical machinery developed in the first half of the talk provides a clear avenue to understand this latter class of algorithms. 
 

Time: November 11, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Wolfgang Dahmen

 Speaker: Jianliang Qian (Michigan State University)

We propose a novel Hadamard-Babich ansatz consisting of an infinite series of dyadic coefficients (three-by-three matrices) and spherical Hankel functions for solving point-source Maxwell's equations in an inhomogeneous medium so as to produce the so-called dyadic Green's function. Using properties of spherical Hankel functions, we derive governing equations for the unknown asymptotics of the ansatz including the travel time function and dyadic coefficients. By proposing matching conditions at the point source, we rigorously derive asymptotic behaviors of these geometrical-optics ingredients near the source so that their initial data at the source point are well-defined. To verify the feasibility of the proposed ansatz, we truncate the ansatz to keep only the first two terms, and we further develop partial-differential-equation based Eulerian approaches to compute the resulting asymptotic solutions. Numerical examples demonstrate that our new ansatz yields a uniform asymptotic solution in the region of space containing a point source but no other caustics. 
 

Time: October 21, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Lili Ju

 Speaker: Zhilin Li (North Carolina State University)

Considering the different backgrounds of the audience, I would like to present an overview of an augmented strategy for solving PDEs hoping to find more applications of the approach. The purpose of the augmented strategy is to decouple some complex systems, rescale or preconditioning PDEs. The augmented strategy makes it possible to obtain accurate and stable discretization. The idea of the augmented strategy for a complicated problem is to introduce some augmented variable(s) along a codimension on a manifold, like a boundary integral method of the source and/or dipole strengths except that no Green's function is needed, more flexible in terms of PDEs (linear or nonlinear), boundary conditions and source terms.

Some important applications will be discussed including the treatment of pressure boundary conditions (not free variables) in Stokes and Navier-Stokes equations; rescaling and fast algorithms for interface problems with large jump ratios, a fluid flow and Darcy's coupling in which the governing equations are different in different regions; and ADI methods for parabolic interface problems, and scattering problems modeled by Maxwell equations, and solver PDEs on irregular domains.  
 

Time: September 30, 2022 2:30pm-3:30pm October 7, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang