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Thursday, 10 January 2019 15:36

On the Euler-alignment system with weakly singular communication weights

 

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
Read 3333 times Last modified on Monday, 19 July 2021 01:09