Alexander Kiselev, and Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 50, No 6, pp. 6208–6229 (2018).

Abstract

The Euler–Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attrac- tion/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in [Do et al., Arch. Ration. Mech. Anal., 228 (2018), pp. 1–37]. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.

	doi:10.1137/17M1141515
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	This work is supported by NSF grant DMS #1853001

Alina Chertock, Changhui Tan, and Bokai Yan

Kinetic and Related Models, Volume 11, No 4, pp. 735-756 (2018).

Abstract

We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors, and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.

	doi:10.3934/krm.2018030
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Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan

Archive for Rational Mechanics and Analysis, Volume 228, No 1, pp. 1-37 (2018).

Abstract

We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian \((-\partial_{xx})^{\alpha/2}, \alpha\in(0,1)\). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all \(\alpha\in(0,1)\). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.

	doi:10.1007/s00205-017-1184-2
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This is a supplementary notes for MATH211 ODE and Linear Algebra.

When I tried to manually adjust the axis limits of a graph, Matlab will automatically cut off the part of the graph that is beyond the axis limits. This didn't happen before (and before means 2014a or earlier versions). I took a little while to find the cure. Just try the following scripts. 

ax = gca;
ax.Clipping = 'off';

Here is the steps to change default mail app in OS X:

1. Open Mail app.
2. Select Preferences -> General. Under "Default email reader", select your preferred mail app.

I am running into a problem by following the steps: the default email reader changes back to the original mail app seconds after I made the change. Here is the solution that works for me. Open terminal. Copy and paste the following script.

/System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/lsregister -kill -r -all local,system,user

Then try step 1-2 again, and it won't change back again!

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.

Download the Undergraduate Thesis (In Chinese)

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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