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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.

### Critical thresholds in flocking hydrodynamics with nonlocal alignment

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372.2028 (2014): 20130401.

Abstract

We study the large-time behavior of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g. Cucker & Smale (2007 IEEE Trans. Autom. Control 52, 852–862. (doi:10.1109/TAC.2007.895842)) and Motsch & Tadmor (2011 J. Stat. Phys. 144, 923–947. (doi:10.1007/s10955-011-0285-9)) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces strong solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.

### Hierarchical construction of bounded solutions to divU = F

"Nonlinear Partial Differential Equations", Proceedings of the 2010 Abel Symposium held in Oslo, Sep. 2010 (H. Holden & K. Karlsen eds.), Abel Symposia 7, Springer 2011, 255-269.

Abstract

We implement the hierarchical decomposition introduced in [Ta15], to construct uniformly bounded solutions of the problem $$\nabla\cdot U = F$$, where the two-dimensional data is in the critical regularity space, $$F\in L^2_{\#}(\mathbb{T}^2)$$. Criticality in this context, manifests itself by the lack of linear mapping, $$F\in L^2_{\#}(\mathbb{T}^2)\to U\in L^{\infty}(\mathbb{T}^2,\mathbb{R}^2)$$ [BB03]. Thus, the intriguing aspect here is that although the problem is linear, the construction of its uniformly bounded solutions is not.

### Application of natural neighbor interpolation method in three-dimensional geological model

Yan-jun Guo, Mao Pan, Fei Yan, Zhe Wang, Changhui Tan, and Tiao Lu

Journal of PLA University of Science and Technology (Natural Science Edition), Volume 26, No 1, pp. 185-206 (2016).

Abstract

To enhance the accuracy of three-dimensional geological model, emphasize the high local relevance characteristics of the complex geological bodies, and avoid complicated calculation and dependence on human experience in traditional interpolation methods, the natural neighbor interpolation (NNI) method was used for three-dimensional discrete data interpolation in the process of modeling. But the existing NNI method could not be applied to the boundary interpolation of finite fields, which was the most difficult problem of its application in three-dimensional geological modeling. Based on the geometry of Voronoi Cells and Delaunay Triangles, the shape function was constructed using non-Sibsonian (Laplace) interpolation method. The continuity of the boundary in NNI method was proven, the boundary interpolation was implemented and the computational complexity was reduced. The accuracy and validity of the method were proven by building the city geological model.

The paper is related to the undergraduate thesis: Numerical analysis and algorithm design in natural neighbor method.

### Honors and Awards

 2013-2014 Ann G. Wylie Dissertation Fellowship SIAM Student Travel Award AMS Graduate Student Travel Grant Kaplan Travel Grant 2011-2012 Mark E. Lachtman Graduate Student Award Jacob K. Goldhaber Travel Award 2010-2011 Kaplan Travel Grant 2008-2010 Graduate Fellowship in University of Maryland 2007-2008 First Honor Graduates in Beijing "Outstanding University Graduates" in Peking University 2006-2007 Triple-Good Student in Peking University Chinese Economical Research Scholarship 2005-2006 Triple-Good Student in Peking University Baogang Scholarship 2004-2005 Triple-Good Student in Peking University Guanghua Scholarship

## 2021

 2021.5.4 PDE/Analysis Seminar, BICRM, Peking University. Talk: Eulerian dynamics in multi-dimensions with radial symmetry. 2021.4.19 Analysis and Applied Mathematics Seminar, University of Illinois Chicago. Talk: The flow of polynomial roots under differentiation. 2021.3.30 Applied Math & Analysis Seminar, Duke University. Talk: Nonlocal traffic flow models and the prevention of traffic jams. 2021.3.26 Zu Chongzhi Colloquium, Duke Kunshan University. Talk: The flow of polynomial roots under differentiation.

## 2020

 2020.11.19 INS Seminar, Shanghai Jiao Tong University. Talk: Nonlocal traffic flow models and the prevention of traffic jams. 2020.10.28 Mathematics Colloquium, Old Dominion University. Talk: Self-organized dynamics: aggregation and flocking. 2020.9.28 CAM Seminar, Iowa State University. Talk: Nonlocal traffic flow models.

## 2019

 2019.12.26 PDE Seminar, Beijing Normal University. Talk: Eulerian dynamics in multi-dimensions with radial symmetry. 2019.12.16-20 Emergent Phenomena – From Kinetic Models to Social Hydrodynamics, National University of Singapore. Talk: Eulerian dynamics in multi-dimensions with radial symmetry. 2019.12.11-14 SIAM Conference on Analysis of Partial Differential Equations: Mini-symposium on Nonlocal PDEs in Fluid Dynamics.[Co-organizer] Talk: Eulerian dynamics in multi-dimensions with radial symmetry. 2019.11.22 Analysis Seminar, University of South Carolina. Talk: Nonlocal traffic flow models. 2019.11.2-3 AMS Sectional Meeting: Special session on Nonlinear PDEs in Fluid Dynamics, University of Florida. Talk: On the Burgers equation with density dependent fractional dissipation. 2019.10.21-25 Young Researchers Workshop: Ki-Net 2012-2019, University of Maryland, College Park. Talk: Eulerian dynamics in multi-dimensions with radial symmetry. 2019.9.20-22 SIAM SEAS 2019 Annual Meeting: Mini-symposium on Recent Developments in Nonlocal PDEs in Fluids and Other Applications, University of Tennessee, Knoxville.[Organizer] 2019.9.20-22 SIAM SEAS 2019 Annual Meeting: Mini-symposium on Innovations and Implementations of Numerical Methods for Time Dependent Problems, University of Tennessee, Knoxville. Talk: Asymptotic preserving schemes on kinetic models with singular limits. 2019.8.19-23 Recent Advances in Nonlocal Kinetic, Fluid and Diffusive PDEs, Jeju, South Korea. Talk: Eulerian dynamics with alignment interactions. 2019.8.13 Analysis Seminar, Korea Institute for Advanced Study, Seoul, South Korea. Talk: Small scale singularity formations in Euler and related equations. 2019.7.15-19 International Congress on Industrial and Applied Mathematics, Valencia, Spain. Talk: Eulerian dynamics with alignment interactions. 2019.7.8-10 2019 TIANFU International Conference on Partial Differential Equations, Southwestern University of Finance and Economics, Chengdu, China. Talk: Eulerian dynamics with alignment interactions. 2019.6.20 PDE Seminar, University of Electronic Science and Technology of China, Chengdu, China. Talk: Small scale singularity formations in Euler and related equations. 2019.6.12 PDE Seminar, Chongqing University, Chongqing, China. Talk: Small scale singularity formations in Euler and related equations. 2019.5.24 Colloquium, Renmin University, Beijing, China. Talk: Small scale singularity formations in Euler and related equations. 2019.5.23 Computational and Applied Math Seminar, Peking University, Beijing, China. Talk: Asymptotic preserving schemes on kinetic models with singular limits. 2019.5.22 Thematic Reports, Beijing Normal University, Beijing, China. Talk: Eulerian dynamics with nonlocal interactions. 2019.5.15 Analysis Seminar, Tsinghua University, Beijing, China. Talk: Self-organized dynamics: aggregation and flocking. 2019.5.9-12 Partial Differential Equations and Applications in Physics and Life Sciences, Chongqing, China. Talk: Eulerian dynamics with alignment interactions. 2019.3.17-20 DASIV Center Spring School Series: Models and Data. Talk: Asymptotic preserving schemes on kinetic models with singular limits. 2019.3.12 PDE Seminar, Georgia Institute of Technology. Talk: Eulerian dynamics with alignment interactions. 2019.3.5-6 Mathematical Aspects of Collective Dynamics: Kinetic Description and Fractional Diffusion, University of Maryland, College Park. Talk: On Euler Alignment system with weakly singular interactions. 2019.3.1 Graduate Seminar, Department of Civil Engineering, University of South Carolina. Talk: The mathematics in continuum mechanics. 2019.2.19 Graduate Colloquium, University of South Carolina. Talk: Fluid dynamics: from stirring a cup of coffee to tracking a hurricane. 2019.2.1 Analysis Seminar, University of South Carolina. Talk: The incompressible Euler equation.

## 2018

 2018.10.22-26 Young Researchers Workshop: Kinetic Descriptions in Theory and Applications, University of Maryland, College Park. Talk: On the Euler-Alignment system. 2018.10.5-7 The 1st Annual Meeting of SIAM Texas-Louisiana Section, Louisiana State University. Talk: Global regularity for Burgers equation with density dependent fractional dissipation. 2018.9.29-30 AMS Sectional Meeting: Special Session on Recent Analytic and Numeric Results on Nonlinear Evolution Equations, University of Delaware. Talk: Global regularity for Burgers equation with density dependent fractional dissipation. 2018.6.18-22 Nonlocal Differential Equations in Collective Behavior, American Institute of Mathematics, San Jose. 2018.4.23 Computational and Applied Mathematics Seminar, Iowa State University. Talk: Self-organized dynamics: aggregation and flocking. 2018.4.21-22 Research Interaction Weekend: Critical Thresholds in Kinetic and Wave Equations, Iowa State University. Talk: Recent progress in critical thresholds phenomenon. 2018.3.1 Mathematics Colloquium, University of Akron. Talk: Self-organized dynamics: aggregation and flocking. 2018.2.26 Mathematics Colloquium, University of Central Florida. Talk: Self-organized dynamics: aggregation and flocking. 2018.2.19-22 Young Researchers Workshop: Kinetic Models in Biology and Social Sciences, Arizona State University. Talk: Asymptotic preserving schemes on kinetic models with singular limits. 2018.2.8 Mathematics Colloquium, University of South Carolina. Talk: Self-organized dynamics: aggregation and flocking. 2018.1.10-13 Joint Mathematics Meeting: AMS Special Session on Analysis of Nonlinear Partial Differential Equations and Applications, San Diego. Talk: Global regularity for Burgers equation with density dependent fractional dissipation. 2018.1.4 Mathematics Colloquium, Baylor University. Talk: Self-organized dynamics: aggregation and flocking.

## 2017

 2017.12.9-12 SIAM Conference on Analysis of Partial Differential Equations, Baltimore. Talk: Global regularity for Burgers equation with density dependent fractional dissipation. 2017.11.30 Hyperbolic and Mixed Type PDEs Seminar, Pennsylvania State University. Talk: Self-organized dynamics: aggregation and flocking. 2017.11.4-5 AMS Sectional Meeting: Special session on Conservation Laws, Nonlinear Waves and Applications, University of California, Riverside. Talk: Global regularity for Burgers equation with density dependent fractional dissipation. 2017.11.4-5 AMS Sectional Meeting: Special session on Mathematical Fluid Mechanics, Nonlinear Waves and Applications, University of California, Riverside. Talk: Regularity and singularity formations for fractional porous medium flow. 2017.11.3 Special Analysis Seminar, University of Southern California. Talk: Self-organized dynamics: aggregation and flocking. 2017.10.26 Differential Equations Seminar, North Carolina State University. Talk: Self-organized dynamics: aggregation and flocking. 2017.10.25 Applied Mathematics and Analysis Seminar, Duke University. Talk: Self-organized dynamics: aggregation and flocking. 2017.10.20 Applied and Computational Mathematics Seminar, Tulane University. Talk: Self-organized dynamics: aggregation and flocking. 2017.10.9-13 Young Researchers Workshop: Current trends in kinetic theory, University of Maryland, College Park. Talk: Kinetic swarming models and hydrodynamic limits. 2017.9.9-10 AMS Sectional Meeting: Special session on Nonlocal PDEs in Fluid Dynamics, University of North Texas, Denton. [Co-organizer] 2017.9.9-10 AMS Sectional Meeting: Special session on Recent Progress on Hyperbolic Conservation Laws, University of North Texas, Denton. Talk: Global regularity for compressible Euler equations with alignment. 2017.7.10-14 SIAM Annual Meeting, Special Session on Recent Development on Fractional Diffusion Equation: Analysis and Computation, Pittsburgh. Talk: Global regularity for the fractional Euler alignment system. 2017.6.5-9 Summer School and Workshop: Mathematical Analysis of Water Waves and Related Models, Bodega Marine Laboratory, Bodega Bay. 2017.4.22-23 AMS Sectional Meeting: Special session on Analysis on the Navier-Stokes Equations and Related PDEs, Washington State University, Pullman. Talk: Global regularity for the fractional Euler alignment system. 2017.4.14 Partial Differential Equations Seminar, University of Houston. Talk: Self-organized dynamics: aggregation and flocking. 2017.3.23-25 Kinetic Descriptions of Chemical and Biological Systems: Models, Analysis and Numerics, Iowa State University, Ames. Talk: Global regularity for the fractional Euler alignment system. 2017.3.4-5 Texas Differential Equations Conference, Texas A&M University, College Station. Talk: Global regularity for Burgers equation with density dependent fractional dissipation. 2017.2.13-17 Dynamics of Small Scales in Fluids, Brown University, Providence.