Alexander Kiselev, and Changhui Tan

Advances in Mathematics, Volume 325, pp. 34-55 (2018).

Abstract

In recent work of Luo and Hou, a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot–Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

	doi:10.1016/j.aim.2017.11.019
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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.

	doi:10.3969/j.issn.1009-3443.2009.06.026
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The paper is related to the undergraduate thesis: Numerical analysis and algorithm design in natural neighbor method.

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Eitan Tadmor, and Changhui Tan

"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.

	doi:10.1007/978-3-642-25361-4_14
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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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