Razvan C. Fetecau, Weiran Sun, and Changhui Tan

Physica D: Nonlinear Phenomena, Volume 325, pp. 146-163 (2016).

Abstract

We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.

	doi:10.1016/j.physd.2016.03.011
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The purpose of this note is to clarify questions and statements in class, and a step-by-step way to find an LU decomposition by hand.

The purpose of this note is to clarify questions and statements in class, and to provide detailed procedures to find polynomial interpolations.

This note is taken in the PDE discussion group in 2012, on the topic of important spaces in fluid dynamics.

Lecture 2: Carleson measure

This note is taken in the PDE discussion group in 2012, on the topic of important spaces in fluid dynamics.

Lecture 1: An introduction to BMO Space

I have finished my Ph.D. defense today.

Committees

Prof. Eitan Tadmor (Chair/Advisor), Prof. Pierre-Emmanuel Jabin, Prof. Dave Levermore, Prof. Antoine Mellet and Prof. Howard Elman (Dean's representative).

My thesis title is Multi-scale problems on collective dynamics and image processing.

	doi:10.13016/M2WG6T
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Eitan Tadmor, and Changhui Tan

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.

	doi:10.1098/rsta.2013.0401
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Thomas Rey, and Changhui Tan

SIAM Journal on Numerical Analysis, Volume 54, No 2, pp. 641-664 (2016).

Abstract

In this work, we discuss kinetic descriptions of flocking models of the so-called Cucker–Smale [IEEE Trans. Automat. Control, 52 (2007), pp. 852–862] and Motsch–Tadmor [J. Statist. Phys., 144 (2011), pp. 923–947] types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potentials. We introduce a new exact rescaling velocity method, inspired by the recent work [F. Filbet and T. Rey, J. Comput. Phys., 248 (2013) pp. 177–199], allowing us to observe numerically the flocking behavior of the solutions to these equations, without a need of remeshing or taking a very fine grid in the velocity space. To stabilize the exact method, we also introduce a modification of the classical upwind finite volume scheme which preserves the physical properties of the solution, such as momentum conservation.

	doi:10.1137/140993430
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I have recently accepted a 3-year Lovett instructor of Mathematics in Rice University. I will work with Professor Alex Kiselev.