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## Research

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

Here is the Curriculum Vitae and List of Publications.

### The flow of polynomial roots under differentiation

Alexander Kiselev, and Changhui Tan

Abstract

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian $$(-\Delta)^{1/2}$$.

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

### Eulerian dynamics in multi-dimensions with radial symmetry

Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 53, No 3, pp. 3040–3071 (2021).

Abstract

We study the global wellposedness of pressureless Eulerian dynamics in multidimensions, with radially symmetric data. Compared with the one-dimensional system, a major difference in multidimensional Eulerian dynamics is the presence of the spectral gap, which is difficult to control in general. We propose a new pair of scalar quantities that provides significantly better control of the spectral gap. Two applications are presented: (i) the Euler-Poisson equations: we show a sharp threshold condition on initial data that distinguish global regularity and finite time blowup; (ii) the Euler-alignment equations: we show a large subcritical region of initial data that leads to global smooth solutions.

 doi:10.1137/20M1358682 Download the Published Version This work is supported by NSF grant DMS #1853001

### On the global classical solution to compressible Euler system with singular velocity alignment

Li Chen, Changhui Tan, and Lining Tong

Methods and Applications of Analysis, to appear.

Abstract

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.

 This work is supported by NSF grant DMS #1853001

### Global regularity for a 1D Euler-alignment system with misalignment

Qianyun Miao, Changhui Tan, and Liutang Xue

Mathematical Models and Methods in Applied Sciences, Volume 31, No 3, pp. 473-524 (2021).

Abstract

We study one-dimensional Eulerian dynamics with nonlocal alignment interactions, fea- turing strong short-range alignment, and long-range misalignment. Compared with the well-studied Euler-alignment system, the presence of the misalignment brings different behaviors of the solutions, including the possible creation of vacuum at infinite time, which destabilizes the solutions. We show that with a strongly singular short-range alignment interaction, the solution is globally regular, despite the effect of misalignment.

 doi:10.1142/S021820252150010X Download the Published Version This work is supported by NSF grant DMS #1853001

### UofSC ASPIRE I Grant: Multiscale nonlocal models in traffic flows

I have been awarded a grant from the Office of the Vice President for Research at the University of South Carolina, on a one-year project: Multiscale nonlocal models in traffic flows.

Project Summary

Mathematical models on traffic flows have been studied extensively in the past century. Many celebrated models lie in a beautiful multiscale framework. The investigations on these models play an important role in designing traffic networks and preventing traffic jams.

Recently, the fast development of self-driving vehicles and new communication technologies allow long-range interactions in traffic networks. It attracts a lot of interest in nonlocal traffic flow models.

The aim of the proposed project is to develop the mathematical theory on non-local traffic flows and understand how nonlocal interactions can help to optimize the traffic networks and avoid the creation of traffic congestions.

The PI has been actively working on a variety of multiscale nonlocal models in physical, biological, and sociological contexts. These experiences can greatly help the understanding of the nonlocal phenomena in traffic models. Preliminary investigations show intriguing behaviors and promising outcomes. Results generated from this project will be capitalized to prepare proposals for external grants from NSF and DOT.