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Comptes Rendus Mathématique
Volume 355, n° 7
pages 795-805 (juillet 2017)
Doi : 10.1016/j.crma.2017.05.008
Received : 23 February 2017 ;  accepted : 29 May 2017
Global regularity of two-dimensional flocking hydrodynamics
Régularité globale des dynamiques d'alignement bidimensionnel à l'échelle hydrodynamique
 

Siming He a , Eitan Tadmor a, b, 1
a Department of Mathematics, Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, USA 
b Institute for Physical Sciences & Technology (IPST), University of Maryland, College Park, MD, USA 

Abstract

We study the systems of Euler equations that arise from agent-based dynamics driven by velocity alignment . It is known that smooth solutions to such systems must flock, namely the large-time behavior of the velocity field approaches a limiting “flocking” velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration that characterizes the global regularity and hence the flocking behavior of such two-dimensional systems. Specifically, we prove for that a large class of sub-critical initial conditions such that the initial divergence is “not too negative” and the initial spectral gap is “not too large”, global regularity persists for all time.

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Résumé

Nous étudions les systémes des équations d'Euler qui résultent de dynamiques d'alignement entre agents. Il a été prouvé que, pour des solutions régulières de tels systémes, en temps grand, le champ de vitesse s'approche d'une vitesse limite uniforme. Nous identifions des seuils critiques dans l'espace de phase de la configuration initiale qui caractérisent la régularité globale et donc le comportement en temps grand de tels systèmes bidimensionnels . Plus précisément, nous prouvons que, pour une classe assez large de conditions initiales sous-critiques telles que la divergence initiale n'est « pas trop négative » et l'écart spectral initial n'est « pas trop grand », la régularité globale reste vraie en temps grand.

The full text of this article is available in PDF format.
1  Current address: ETH Institute for Theoretical Studies, ETH-Zürich, 8092 Zürich, Switzerland.
2  We let   denote the usual   norm.
3  Equating the trace of   with that of   we find  . Using   with   on the left and   on the right implies ((2.11)).
4  The use of such scaling simplifies the computation below.


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