Global regularity of two-dimensional flocking hydrodynamics

Siming He; Eitan Tadmor

doi:10.1016/j.crma.2017.05.008

Partial differential equations

Global regularity of two-dimensional flocking hydrodynamics
[Régularité globale des dynamiques d'alignement bidimensionnel à l'échelle hydrodynamique]

Présenté par : Jean-Michel Coron

Siming He ¹ ; Eitan Tadmor ^1,²

¹ Department of Mathematics, Center for Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD, USA
² Institute for Physical Sciences & Technology (IPST), University of Maryland, College Park, MD, USA

Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 795-805.

Résumé
Abstract

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.

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.

Reçu le : 2017-02-23
Accepté le : 2017-05-29
Publié le : 2017-06-16

DOI : 10.1016/j.crma.2017.05.008

Affiliations des auteurs :

Siming He ¹ ; Eitan Tadmor ^{1, 2}

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     title = {Global regularity of two-dimensional flocking hydrodynamics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {795--805},
     publisher = {Elsevier},
     volume = {355},
     number = {7},
     year = {2017},
     doi = {10.1016/j.crma.2017.05.008},
     language = {en},
}

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Siming He; Eitan Tadmor. Global regularity of two-dimensional flocking hydrodynamics. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 795-805. doi : 10.1016/j.crma.2017.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.008/

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