Methods and Applications of Analysis

Volume 28 (2021)

Number 2

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part I

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

On the global classical solution to compressible Euler system with singular velocity alignment

Pages: 153 – 172



Li Chen (Department of Mathematics, University of Mannheim, Germany)

Changhui Tan (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Lining Tong (Department of Mathematics, Shanghai University, Shanghai, China)


We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.


Euler-alignment system, singular velocity alignment, global existence

2010 Mathematics Subject Classification

35L65, 35Q35, 35Q70

The research of Changhui Tan was supported by NSF grant DMS 1853001.

The research Lining Tong was supported by NSFC grants 11771274 and 11901379.

Received 16 July 2020

Accepted 7 October 2020

Published 10 June 2022