-
Previous Article
Stochastic functional Hamiltonian system with singular coefficients
- CPAA Home
- This Issue
-
Next Article
Bifurcation and stability of a two-species diffusive Lotka-Volterra model
Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime
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 |
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 [
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. |
[1] |
Matthias Erbar, Dominik Forkert, Jan Maas, Delio Mugnolo. Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022023 |
[2] |
Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic and Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415 |
[3] |
Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic and Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010 |
[4] |
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 |
[5] |
Christian Bourdarias, Marguerite Gisclon, Stéphane Junca. Kinetic formulation of a 2 × 2 hyperbolic system arising in gas chromatography. Kinetic and Related Models, 2020, 13 (5) : 869-888. doi: 10.3934/krm.2020030 |
[6] |
Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic and Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901 |
[7] |
Haiyang Wang, Jianfeng Zhang. Forward backward SDEs in weak formulation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1021-1049. doi: 10.3934/mcrf.2018044 |
[8] |
François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41 |
[9] |
Matthew M. Dunlop, Andrew M. Stuart. The Bayesian formulation of EIT: Analysis and algorithms. Inverse Problems and Imaging, 2016, 10 (4) : 1007-1036. doi: 10.3934/ipi.2016030 |
[10] |
Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917 |
[11] |
André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135 |
[12] |
Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337 |
[13] |
Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845 |
[14] |
Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587 |
[15] |
Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343 |
[16] |
Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373 |
[17] |
Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023 |
[18] |
Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419 |
[19] |
Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 |
[20] |
Xiaoliang Cheng, Rongfang Gong, Weimin Han. A new Kohn-Vogelius type formulation for inverse source problems. Inverse Problems and Imaging, 2015, 9 (4) : 1051-1067. doi: 10.3934/ipi.2015.9.1051 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]