Eulerian dynamics with a commutator forcing Ⅱ: Flocking

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

Eulerian dynamics with a commutator forcing Ⅱ: Flocking

Roman Shvydkoy ^1, and Eitan Tadmor ^2,3,,

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 $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.

Keywords: Flocking, alignment, fractional dissipation, Cucker-Smale, Motsch-Tadmor, critical thresholds.

Mathematics Subject Classification: Primary: 35Q35, 76N10; Secondary: 92D25.

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

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.

show all references

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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