Research

Research

Here are the latest updates for Changhui Tan's research profile.

Here is the Curriculum Vitae and List of Publications.

Monday, 13 May 2019 15:36

A sharp critical threshold for a traffic flow model with look-ahead dynamics

Yongki Lee, and Changhui Tan

Abstract

We study a Lighthill-Whitham-Richards (LWR) type traffic flow model, with an Arrhenius type look-ahead interaction. We show a sharp critical threshold condition on the initial data which distinguishes the global smooth solutions and finite time wave break-down. It is well-known that the LWR model leads to a finite time shock formation, representing traffic jams, for any nontrivial initial data. Our result shows that the nonlocal slowdown effect can help to avoid traffic jams, for a class of subcritical initial data.

 This work is supported by NSF grant DMS #1853001 This work is supported by a UofSC VPR ASPIRE I grant

Thursday, 10 January 2019 15:36

On the Euler-alignment system with weakly singular communication weights

Changhui Tan

Nonlinearity, Volume 33, No 4, pp. 1907-1924 (2020).

Abstract

We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of the Cucker–Smale model on animal flocks. For the Euler-alignment system with bounded interactions, a critical threshold phenomenon is proved in Tadmor and Tan (2014 Phil. Trans. R. Soc. A 372 20130401), where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in Do et al (2018 Arch. Ration. Mech. Anal. 228 1–37), for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.

 doi:10.1088/1361-6544/ab6c39 Download the Published Version This work is supported by NSF grant DMS #1853001
Tuesday, 24 July 2018 00:59

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.

Wednesday, 30 August 2017 22:56

Singularity formation for a fluid mechanics model with nonlocal velocity

Changhui Tan

Communications in Mathematical Sciences, Volume 17, No 7, pp. 1779-1794 (2019).

Abstract

We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of quasi-geostrophic equation, and also a special case of the Euler alignment system. For strictly positive smooth initial data, global regularity has been proved in [Do, Kiselev, Ryzhik and Tan, Arch. Ration. Mech. Anal., 228(1):1–37, 2018]. We construct a family of non-negative smooth initial data so that solution is not $$C^1$$-uniformly bounded. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.

 doi:10.4310/CMS.2019.v17.n7.a2 Download the Published Version This work is supported by NSF grant DMS #1853001
Monday, 24 July 2017 23:04

Global regularity for 1D Eulerian dynamics with singular interaction forces

Alexander Kiselev, and Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 50, No 6, pp. 6208–6229 (2018).

Abstract

The Euler–Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attrac- tion/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in [Do et al., Arch. Ration. Mech. Anal., 228 (2018), pp. 1–37]. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.

 doi:10.1137/17M1141515 Download the Published Version This work is supported by NSF grant DMS #1853001
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