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
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	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

Xiang Bai, Qianyun Miao, Changhui Tan and Liutang Xue

Nonlinearity, Volume 37, 025007, 46pp. (2024).

Abstract

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as \(t\to\infty\).

	doi:10.1088/1361-6544/ad140b
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

Yi Sun and Changhui Tan

Physica D, Volume 446, 133657 (2023)

Abstract

This paper presents a class of one-dimensional cellular automata (CA) models on traffic flows, featuring nonlocal look-ahead interactions. We develop kinetic Monte Carlo (KMC) algorithms to simulate the dynamics. The standard KMC method can be inefficient for models with global interactions. We design an accelerated KMC method to reduce the computational complexity in the evaluation of the nonlocal transition rates. We investigate several numerical experiments to demonstrate the efficiency of the accelerated algorithm, and obtain the fundamental diagrams of the dynamics under various parameter settings.

	doi:10.1016/j.physd.2023.133657
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

 Speaker: Ziad Musslimani (Florida State University)

In this talk we shall outline a new method to solve initial and boundary value problems of physical relevance. The idea is to use the underlying physics (such as conservation laws or dissipation rate equations) combined with a dynamic renormalization process to numerically compute ground and excited states as well as time-dependent solutions. We will apply the method on a prototypical problems that arise in physics such as Gross-Pitaevski equation and Hartree-Fock.
 

Time: April 22, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang