American Institute of Mathematical Sciences

Advanced
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[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.  Google Scholar

[11]

R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017), 1-26.  doi: 10.1093/imatrm/tnx001.  Google Scholar

[12]

R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[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.  Google Scholar

[11]

R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017), 1-26.  doi: 10.1093/imatrm/tnx001.  Google Scholar

[12]

R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246. Google Scholar

[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.  Google Scholar

[1]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
[2]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168
[3]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[4]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419
[5]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062
[6]

Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic & Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008
[7]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072
[8]

Moon-Jin Kang, Seung-Yeal Ha, Jeongho Kim, Woojoo Shim. Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1233-1256. doi: 10.3934/cpaa.2020057
[9]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116
[10]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028
[11]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[12]

Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic & Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027
[13]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020155
[14]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic & Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026
[15]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[16]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028
[17]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[18]

Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447
[19]

Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115
[20]

Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (65)
  • HTML views (68)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences