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March  2020, 19(3): 1233-1256. doi: 10.3934/cpaa.2020057

Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime

Moon-Jin Kang 1, , Seung-Yeal Ha 2, , Jeongho Kim 3, and  Woojoo Shim 4,,

1. 

Department of Mathematics and Research Institute of Natural Sciences, Sookmyung Women's University, Seoul 04310, Republic of Korea
2. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
3. 

Institute of New Media and Communications, Seoul National University, Seoul 08826, Republic of Korea
4. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author

Received  October 2018 Revised  August 2019 Published  November 2019

We present a hydrodynamic limit from the kinetic thermomechanical Cucker-Smale (TCS) model to the hydrodynamic Cucker-Smale (CS) model in a strong local alignment regime. For this, we first provide a global existence of weak solution, and flocking dynamics for classical solution to the kinetic TCS model with local alignment force. Then we consider one-parameter family of well-prepared initial data to the kinetic TCS model in which the temperature tends to common constant value determined by initial datum, as singular parameter $ \varepsilon $ tends to zero. In a strong local alignment regime, the limit model is the hydrodynamic CS model in [8]. To verify this hydrodynamic limit rigorously, we adopt the technique introduced in [5] which combines the relative entropy method together with the 2-Wasserstein metric.

Keywords: Flocking, particles, kinetic formulation, hydrodynamic formulation, Wasserstein metric.
Mathematics Subject Classification: Primary: 92D25, 74A25; Secondary: 76N10.
Citation: 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 and Applied Analysis, 2020, 19 (3) : 1233-1256. doi: 10.3934/cpaa.2020057
References:
[1]

Y.-P. Choi, S.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[2]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.

[3]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, To appear in J. Math. Fluid Mech.

[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]

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.

[6]

S.-Y. Ha, M.-J. Kang and B. Kwon, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Models Methods Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.

[7]

S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[8]

S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, A global existence of classical solution to the hydrodynamic Cucker-Smale model in presence of temperature field, Anal. Appl., 16 (2018), 757-805.  doi: 10.1142/S0219530518500033.

[9]

S.-Y. Ha, J. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.

[10]

S.-Y. Ha, Z. Li, M. Slemrod and X. Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math., 72 (2014), 689-701.  doi: 10.1090/S0033-569X-2014-01350-5.

[11]

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. 

[12]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Rational Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.

[13]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[14]

P.-E. Jabin and T. Rey, Hydrodynamic limit of granular gases to pressureless Euler in dimension one, Quart. Appl. Math., 75 (2017), 155-179.  doi: 10.1090/qam/1442.

[15]

M.-J. Kang, From the Vlasov-Poisson equation with strong local alignment to the pressureless Euler-Poisson system, Appl. Math. Lett., 79 (2018), 85-91.  doi: 10.1016/j.aml.2017.12.001.

[16]

M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.

[17]

T. K. Karper, A. 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.

[18]

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

[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, in Springer Proceedings in Math. Statistics, 49 (2014), 227-242. doi: 10.1007/978-3-642-39007-4_11.

[20]

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

[21]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 141 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[22]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089-1152.  doi: 10.1142/S0218202517400103.

[23]

A. Vasseur, Recent results on hydrodynamic limits, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, in: Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 323–376. doi: 10.1016/S1874-5717(08)00007-8.

[24]

T. Vicsek, Cz irók, E. Ben-Jacob, I. Cohen and O. Schochet, 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.

show all references

References:
[1]

Y.-P. Choi, S.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[2]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.  doi: 10.1088/1361-6544/aafaae.

[3]

Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim, On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, To appear in J. Math. Fluid Mech.

[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]

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.

[6]

S.-Y. Ha, M.-J. Kang and B. Kwon, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Models Methods Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.

[7]

S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.  doi: 10.1090/qam/1517.

[8]

S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang, A global existence of classical solution to the hydrodynamic Cucker-Smale model in presence of temperature field, Anal. Appl., 16 (2018), 757-805.  doi: 10.1142/S0219530518500033.

[9]

S.-Y. Ha, J. Kim and T. Ruggeri, Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.  doi: 10.1137/17M111064X.

[10]

S.-Y. Ha, Z. Li, M. Slemrod and X. Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math., 72 (2014), 689-701.  doi: 10.1090/S0033-569X-2014-01350-5.

[11]

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. 

[12]

S.-Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Rational Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.

[13]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[14]

P.-E. Jabin and T. Rey, Hydrodynamic limit of granular gases to pressureless Euler in dimension one, Quart. Appl. Math., 75 (2017), 155-179.  doi: 10.1090/qam/1442.

[15]

M.-J. Kang, From the Vlasov-Poisson equation with strong local alignment to the pressureless Euler-Poisson system, Appl. Math. Lett., 79 (2018), 85-91.  doi: 10.1016/j.aml.2017.12.001.

[16]

M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.

[17]

T. K. Karper, A. 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.

[18]

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

[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, in Springer Proceedings in Math. Statistics, 49 (2014), 227-242. doi: 10.1007/978-3-642-39007-4_11.

[20]

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

[21]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 141 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.

[22]

D. Poyato and J. Soler, Euler-type equations and commutators in singular and hyperbolic limits of kinetic Cucker-Smale models, Math. Models Methods Appl. Sci., 27 (2017), 1089-1152.  doi: 10.1142/S0218202517400103.

[23]

A. Vasseur, Recent results on hydrodynamic limits, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, in: Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 323–376. doi: 10.1016/S1874-5717(08)00007-8.

[24]

T. Vicsek, Cz irók, E. Ben-Jacob, I. Cohen and O. Schochet, 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.

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