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. 

Jose A. Carrillo, Young-Pil Choi, Eitan Tadmor, and Changhui Tan

Mathematical Models and Methods in Applied Sciences, Volume 26, No 1, pp. 185-206 (2016).

Abstract

We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global-in-time existence or finite-time blowup of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also remark on global-in-time existence when the repulsion is modeled by the isothermal pressure law.

	doi:10.1142/S0218202516500068
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Changhui Tan

Mathematical Models and Methods in Applied Sciences, Volume 27, No 7, pp. 1199-1221 (2017).

Abstract

We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker–Smale [Emergent behaviors in flocks, IEEE Trans. Autom. Control. 52 (2007) 852–862] and Motsch–Tadmor [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 144 (2011) 923– 947] models. We first establish a well-posedness theory and large-time flocking behavior for the kinetic systems, which indicates a concentration in velocity variable in infinite time. We then apply a discontinuous Galerkin method to treat the asymptotic \(\delta\)-singularity, and construct high-order positive-preserving schemes to solve kinetic flocking systems.

	doi:10.1142/S0218202517400139
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