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

 Speaker: Qingtian Zhang (West Virginia University)

In this talk, I will introduce the vortex front problem for quasi-geostrophic shallow water equation, which is also known as Hasegawa-Mima equation in plasma science. The contour dynamic equation of the vortex front will be derived, which is a nonlocal, nonlinear dispersive equation. The existence of global solutions will be proved when the initial data is small.
 

Time: March 25, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Thomas Hamori and Changhui Tan

Nonlinear Analysis: Real World Applications, Volume 73, 103899, (2023).

Abstract

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.

	doi:10.1016/j.nonrwa.2023.103899
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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: Guowei Wei (Michigan State University)

Mathematics underpins fundamental theories in physics such as quantum mechanics, general relativity, and quantum field theory. Nonetheless, its success in modern biology, namely cellular biology, molecular biology, biochemistry, genomics, and genetics, has been quite limited. Artificial intelligence (AI) has fundamentally changed the landscape of science, technology, industry, and social media in the past few years and holds a great future for discovering the rules of life. However, AI-based biological discovery encounters challenges arising from the structural complexity of macromolecules, the high dimensionality of biological variability, the multiscale entanglement of molecules, cells, tissues, organs, and organisms, the nonlinearity of genotype, phenotype, and environment coupling, and the excessiveness of genomic, transcriptomic, proteomic, and metabolomic data. We tackle these challenges mathematically. Our work focuses on reducing the complexity, dimensionality, entanglement, and nonlinearity of biological data in AI. We have introduced evolutionary de Rham-Hodge, persistent cohomology, persistent Laplacian, and persistent sheaf theories to model complex, heterogeneous, multiscale biological systems and thus significantly enhance AI's ability to handle biological datasets. Using our mathematical AI approaches, my team has been the top winner in D3R Grand Challenges, a worldwide annual competition series in computer-aided drug design and discovery for years. Using over two million genomes isolates from patients, we discovered the mechanisms of SARS-CoV-2 evolution and transmission and accurately forecast emerging SARS-CoV-2 variants.
 

Time: April 15, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang

 Speaker: Ming Chen (University of Pittsburgh)

I will discuss the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. We prove that for supercritical surface tension, all known small-amplitude localized waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. For the near critical surface tension regime, we show that one can infer conditional orbital stability or orbital instability of small-amplitude traveling waves solutions to the full Euler system from considerations of a dispersive PDE similar to the steady Kawahara equation. This is joint work with S. Walsh.
 

Time: March 4, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

 Speaker: Jiahong Wu (Oklahoma State University)

The background magnetic field stabilizes and damps electrically conducting fluids, and the temperature tames and stabilizes buoyancy driven fluids. These are just two examples of a seemingly universal stabilizing phenomenon that has been experimentally and numerically observed for different types of incompressible fluids. This talk presents recent work that establishes this phenomenon as mathematically rigorous stability results. In particular, we describe the global existence and stability results for the 3D incompressible anisotropic magnetohydrodynamic system near a background magnetic field, for the Boussinesq system near the hydrostatic equilibrium, and for the Oldroyd-B model near the trivial solution.
 

Time: February 18, 2022 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

Manas Bhatnagar, Hailiang Liu and Changhui Tan

Journal of Differential Equations, Volume 375, pp. 82-119 (2023)

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.

	doi:10.1016/j.jde.2023.07.049
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	This work is supported by NSF grants DMS #1853001, DMS #2108264 and DMS 2238219

 Speaker: Xiaochuan Tian (University of California, San Diego)

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. In this talk, we will give a review of the asymptotically compatible schemes for nonlocal models with a parameter dependence. Such numerical schemes are robust under the change of the nonlocal length parameter and are suitable for multiscale simulations where nonlocal and local models are coupled. We will discuss finite difference, finite element and collocation methods for nonlocal models as well as the related open questions for each type of the numerical methods.
 

Time: November 12, 2021 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Changhui Tan