# American Institute of Mathematical Sciences

November  2017, 37(11): 5503-5520. doi: 10.3934/dcds.2017239

## Eulerian dynamics with a commutator forcing Ⅱ: Flocking

 1 Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois, Chicago, IL 60607, USA 2 Center for Scientific Computation and Mathematical Modeling (CSCAMM), Department of Mathematics, Institute for Physical Sciences and Technology, University of Maryland, College Park, MD 20742-4015, USA 3 Current address: Institute for Theoretical Studies (ITS), ETH-Zurich, Clausiusstrasse 47, CH-8092 Zurich, Switzerland

* Corresponding author: Eitan Tadmor

Received  January 2017 Revised  June 2017 Published  July 2017

Fund Project: Research was supported in part by NSF grant DMS 1515705 (RS) and by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR grant N00014-1512094 (ET)

We continue our study of one-dimensional class of Euler equations, introduced in [11], driven by a forcing with a commutator structure of the form $[{\mathcal L}_φ, u](ρ)$, where $u$ is the velocity field and ${\mathcal L}_φ$ belongs to a rather general class of convolution operators depending on interaction kernels $φ$.

In this paper we quantify the large-time behavior of such systems in terms of fast flocking, for two prototypical sub-classes of kernels: bounded positive $φ$'s, and singular $φ(r) = r^{-(1+α)}$ of order $α∈ [1, 2)$ associated with the action of the fractional Laplacian ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(·, t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.

Citation: Roman Shvydkoy, Eitan Tadmor. Eulerian dynamics with a commutator forcing Ⅱ: Flocking. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5503-5520. doi: 10.3934/dcds.2017239
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