# 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 and Continuous Dynamical Systems, 2017, 37 (11) : 5503-5520. doi: 10.3934/dcds.2017239
##### References:
 [1] J.A. Carrillo, Y.-P. Choi, E. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26 (2016), 185-206.  doi: 10.1142/S0218202516500068. [2] J. Carrillo, Y.-P. Choi and S. Perez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles, Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET), (2017), 259-298.  doi: 10.1007/978-3-319-49996-3_7. [3] P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.  doi: 10.1007/s00039-012-0172-9. [4] T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155. [5] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [6] C. Imbert, R. Shvydkoy and F. Vigneron, Global well-posedness of a non-local Burgers equation: The periodic case, Annales mathématiques de Toulouse, 25 (2016), 723-758.  doi: 10.5802/afst.1509. [7] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2{D} dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.  doi: 10.1007/s00222-006-0020-3. [8] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9. [9] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866. [10] R.W. Schwab and L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, 9 (2016), 727-772.  doi: 10.2140/apde.2016.9.727. [11] R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017), 1-26.  doi: 10.1093/imatrm/tnx001. [12] R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246. [13] E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401, 22pp. doi: 10.1098/rsta.2013.0401.

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##### References:
 [1] J.A. Carrillo, Y.-P. Choi, E. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26 (2016), 185-206.  doi: 10.1142/S0218202516500068. [2] J. Carrillo, Y.-P. Choi and S. Perez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles, Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET), (2017), 259-298.  doi: 10.1007/978-3-319-49996-3_7. [3] P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.  doi: 10.1007/s00039-012-0172-9. [4] T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155. [5] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [6] C. Imbert, R. Shvydkoy and F. Vigneron, Global well-posedness of a non-local Burgers equation: The periodic case, Annales mathématiques de Toulouse, 25 (2016), 723-758.  doi: 10.5802/afst.1509. [7] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2{D} dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.  doi: 10.1007/s00222-006-0020-3. [8] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9. [9] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866. [10] R.W. Schwab and L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, 9 (2016), 727-772.  doi: 10.2140/apde.2016.9.727. [11] R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017), 1-26.  doi: 10.1093/imatrm/tnx001. [12] R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246. [13] E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401, 22pp. doi: 10.1098/rsta.2013.0401.
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