April  2022, 15(2): 213-237. doi: 10.3934/krm.2022005

Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

851 S Morgan St, M/C 249, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

* Corresponding author: Roman Shvydkoy

Received  July 2021 Revised  January 2022 Published  April 2022 Early access  January 2022

Fund Project: This work was supported in part by NSF grants DMS-1813351 and DMS-2107956

In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e. it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation.

The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.

Citation: Roman Shvydkoy. Global hypocoercivity of kinetic Fokker-Planck-Alignment equations. Kinetic and Related Models, 2022, 15 (2) : 213-237. doi: 10.3934/krm.2022005
References:
[1]

F. BolleyJ. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.

[2]

S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations, Comm. Partial Differential Equations, 37 (2012), 1357-1390.  doi: 10.1080/03605302.2011.648039.

[3]

Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887.

[4]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[5]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[6]

H. Dietert and R. Shvydkoy, On Cucker-Smale dynamical systems with degenerate communication, Anal. Appl. (Singap.), 19 (2021), 551-573.  doi: 10.1142/S0219530520500050.

[7]

R. Duan, The Boltzmann equation near equilibrium states in $\Bbb R^N$, Methods Appl. Anal., 14 (2007), 227-249.  doi: 10.4310/MAA.2007.v14.n3.a2.

[8]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[9]

A. Favre, Turbulence: Space-time statistical properties and behavior in supersonic flows, Phys. Fluids, 26 (1983), 2851-2863.  doi: 10.1063/1.864049.

[10]

A. Figalli and M.-J. Kang, A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE, 12 (2019), 843-866.  doi: 10.2140/apde.2019.12.843.

[11]

T. GoudonP.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515.  doi: 10.1512/iumj.2004.53.2508.

[12]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[13]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.

[14]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[15]

J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006.

[16]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis and simulation, J. Artifical Societies Social Simul., 5 (2002). Available from: https://www.math.fsu.edu/ dgalvis/journalclub/papers/02_05_2017.pdf.

[17]

T. K. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM. J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.

[18]

T. K. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[19]

T. K. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, in Hyperbolic Conservation Laws and Related Analysis With Applications, Springer Proc. Math. Stat., 49, Springer, Heidelberg, 2014,227–242. doi: 10.1007/978-3-642-39007-4_11.

[20]

T. M. Leslie and R. Shvydkoy, The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differential Equations, 261 (2016), 3719-3733.  doi: 10.1016/j.jde.2016.06.001.

[21]

T. M. Leslie and R. Shvydkoy, On the structure of limiting flocks in hydrodynamic Euler Alignment models, Math. Models Methods Appl. Sci., 29 (2019), 2419-2431.  doi: 10.1142/S0218202519500507.

[22]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.

[23]

P. MinakowskiP. B. Mucha and J. Peszek, Density-induced consensus protocol, Math. Models Methods Appl. Sci., 30 (2020), 2389-2415.  doi: 10.1142/S0218202520500451.

[24]

P. Minakowski, P. B. Mucha, J. Peszek and E. Zatorska, Singular Cucker-Smale dynamics, in Active Particles, Vol. 2, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2019,201–243.

[25]

J. MoralesJ. Peszek and E. Tadmor, Flocking with short-range interactions, J. Stat. Phys., 176 (2019), 382-397.  doi: 10.1007/s10955-019-02304-5.

[26]

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

[27]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[28]

A. Rosello, Weak and strong mean-field limits for stochastic Cucker-Smale particle systems, preprint, 2019, arXiv: 1905.02499.

[29]

R. Shu and E. Tadmor, Flocking hydrodynamics with external potentials, Arch. Ration. Mech. Anal., 238 (2020), 347-381.  doi: 10.1007/s00205-020-01544-0.

[30]

R. Shvydkoy, Dynamics and Analysis of Alignment Models of Collective Behavior, Nečas Center Series, Birkhäuser/Springer, Cham, 2021. doi: 10.1007/978-3-030-68147-0.

[31]

R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing II: Flocking, Discrete Contin. Dyn. Syst., 37 (2017), 5503-5520.  doi: 10.3934/dcds.2017239.

[32]

R. Shvydkoy and E. Tadmor, Topologically based fractional diffusion and emergent dynamics with short-range interactions, SIAM J. Math. Anal., 52 (2020), 5792-5839.  doi: 10.1137/19M1292412.

[33]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[34]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[35]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

show all references

References:
[1]

F. BolleyJ. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.

[2]

S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations, Comm. Partial Differential Equations, 37 (2012), 1357-1390.  doi: 10.1080/03605302.2011.648039.

[3]

Y.-P. Choi, Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces, Nonlinearity, 29 (2016), 1887-1916.  doi: 10.1088/0951-7715/29/7/1887.

[4]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[5]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[6]

H. Dietert and R. Shvydkoy, On Cucker-Smale dynamical systems with degenerate communication, Anal. Appl. (Singap.), 19 (2021), 551-573.  doi: 10.1142/S0219530520500050.

[7]

R. Duan, The Boltzmann equation near equilibrium states in $\Bbb R^N$, Methods Appl. Anal., 14 (2007), 227-249.  doi: 10.4310/MAA.2007.v14.n3.a2.

[8]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[9]

A. Favre, Turbulence: Space-time statistical properties and behavior in supersonic flows, Phys. Fluids, 26 (1983), 2851-2863.  doi: 10.1063/1.864049.

[10]

A. Figalli and M.-J. Kang, A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment, Anal. PDE, 12 (2019), 843-866.  doi: 10.2140/apde.2019.12.843.

[11]

T. GoudonP.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515.  doi: 10.1512/iumj.2004.53.2508.

[12]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.  doi: 10.2307/2373688.

[13]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.

[14]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.

[15]

J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006.

[16]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis and simulation, J. Artifical Societies Social Simul., 5 (2002). Available from: https://www.math.fsu.edu/ dgalvis/journalclub/papers/02_05_2017.pdf.

[17]

T. K. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM. J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.

[18]

T. K. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale flocking model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.

[19]

T. K. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, in Hyperbolic Conservation Laws and Related Analysis With Applications, Springer Proc. Math. Stat., 49, Springer, Heidelberg, 2014,227–242. doi: 10.1007/978-3-642-39007-4_11.

[20]

T. M. Leslie and R. Shvydkoy, The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differential Equations, 261 (2016), 3719-3733.  doi: 10.1016/j.jde.2016.06.001.

[21]

T. M. Leslie and R. Shvydkoy, On the structure of limiting flocks in hydrodynamic Euler Alignment models, Math. Models Methods Appl. Sci., 29 (2019), 2419-2431.  doi: 10.1142/S0218202519500507.

[22]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.

[23]

P. MinakowskiP. B. Mucha and J. Peszek, Density-induced consensus protocol, Math. Models Methods Appl. Sci., 30 (2020), 2389-2415.  doi: 10.1142/S0218202520500451.

[24]

P. Minakowski, P. B. Mucha, J. Peszek and E. Zatorska, Singular Cucker-Smale dynamics, in Active Particles, Vol. 2, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2019,201–243.

[25]

J. MoralesJ. Peszek and E. Tadmor, Flocking with short-range interactions, J. Stat. Phys., 176 (2019), 382-397.  doi: 10.1007/s10955-019-02304-5.

[26]

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

[27]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[28]

A. Rosello, Weak and strong mean-field limits for stochastic Cucker-Smale particle systems, preprint, 2019, arXiv: 1905.02499.

[29]

R. Shu and E. Tadmor, Flocking hydrodynamics with external potentials, Arch. Ration. Mech. Anal., 238 (2020), 347-381.  doi: 10.1007/s00205-020-01544-0.

[30]

R. Shvydkoy, Dynamics and Analysis of Alignment Models of Collective Behavior, Nečas Center Series, Birkhäuser/Springer, Cham, 2021. doi: 10.1007/978-3-030-68147-0.

[31]

R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing II: Flocking, Discrete Contin. Dyn. Syst., 37 (2017), 5503-5520.  doi: 10.3934/dcds.2017239.

[32]

R. Shvydkoy and E. Tadmor, Topologically based fractional diffusion and emergent dynamics with short-range interactions, SIAM J. Math. Anal., 52 (2020), 5792-5839.  doi: 10.1137/19M1292412.

[33]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.

[34]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[35]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.

[1]

Linglong Du, Xinyun Zhou. The stochastic delayed Cucker-Smale system in a harmonic potential field. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022022

[2]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011

[3]

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

[4]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[5]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016

[6]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485

[7]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[8]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

[9]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008

[10]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028

[11]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[12]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028

[13]

Michael Blank. Emergence of collective behavior in dynamical networks. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 313-329. doi: 10.3934/dcdsb.2013.18.313

[14]

Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks and Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032

[15]

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

[16]

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

[17]

Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195

[18]

Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164

[19]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[20]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (135)
  • HTML views (97)
  • Cited by (0)

Other articles
by authors

[Back to Top]