Speaker: Xiangcheng Zheng (University of South Carolina)

We proved the wellposedness of variable-order mobile-immobile time-fractional diffusion equations and the regularity of their solutions. Optimal-order finite element approximation was presented and analyzed. Numerical experiments were carried out for demonstration.
 

Time: January 31, 2020 2:30pm-3:30pm
Location: LC317R

Speaker: Xiaolin Li (Stony Brook University)

In this talk, I will review the history of the Lagrangian front tracking method and the computational platform built on this methodology. I will review the front tracking in the study of fluid interface instabilities, including Rayleigh-Taylor instability, Richtmyer Meshkov instability and fluid mixing induced by these instabilities. I will also introduce a fully conservative front tracking scheme and its application in the phase transition problem.
 

Time: March 20, 2020 2:30pm-3:30pm
Location: LC317R
Host: Xinfeng Liu

 Speaker: Yuanwei Qi (University of Central Florida)

In this talk, I shall report some recent progress on the existence and multiplicity of traveling waves to one of the most important models in Turing Pattern Formation. In addition, I shall pose some questions which are wide open which demand new ideas and fresh approaches. This is a joint-work with Xinfu Chen et al.
 

Time: January 17, 2020 2:30pm-3:30pm
Location: LC317R
Host: Changhui Tan

In the MacOS system, the keychain password has to match the user password in order to use keychain appropriately. In my case, they have to match in order to allow Apple Watch to unlock the Mac.

The official way to change the keychain password can be found HERE. However, in my case, the "Change password for keychain login" option is in gray and can not be selected. One solution is the following:

Open terminal and type in

security set-keychain-password

Then, type in old password and then new password. The password is then changed. This does not require admin access.

Yongki Lee, and Changhui Tan

Communications in Mathematical Sciences, Volume 20, No. 4, pp. 1151-1172 (2022).

Abstract

We study a Lighthill-Whitham-Richards (LWR) type traffic flow model, with a nonlocal look-ahead interaction that has a slow-down effect depending on the traffic ahead. We show a sharp critical threshold condition on the initial data that distinguishes global smooth solutions and finite- time wave breakdown. It is well-known that the LWR model leads to a finite-time shock formation, representing the creation of traffic jams, for generic smooth initial data with finite mass. Our result shows that the nonlocal slowdown effect can help to prevent shock formations, for a class of subcritical initial data.

	doi:10.4310/CMS.2022.v20.n4.a9
	Download the Published Version
	This work is supported by NSF grants DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

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
	Download the Published Version
	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
	Download the Published Version
	This work is supported by NSF grant DMS #1853001