Changhui Tan

Nonlinearity, Volume 33, No 4, pp. 1907-1924 (2020).

Abstract

We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of the Cucker–Smale model on animal flocks. For the Euler-alignment system with bounded interactions, a critical threshold phenomenon is proved in Tadmor and Tan (2014 Phil. Trans. R. Soc. A 372 20130401), where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in Do et al (2018 Arch. Ration. Mech. Anal. 228 1–37), for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.

	doi:10.1088/1361-6544/ab6c39
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	This work is supported by NSF grant DMS #1853001

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Changhui Tan

Communications in Mathematical Sciences, Volume 17, No 7, pp. 1779-1794 (2019).

Abstract

We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of quasi-geostrophic equation, and also a special case of the Euler alignment system. For strictly positive smooth initial data, global regularity has been proved in [Do, Kiselev, Ryzhik and Tan, Arch. Ration. Mech. Anal., 228(1):1–37, 2018]. We construct a family of non-negative smooth initial data so that solution is not \(C^1\)-uniformly bounded. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.

	doi:10.4310/CMS.2019.v17.n7.a2
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	This work is supported by NSF grant DMS #1853001

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!