American Institute of Mathematical Sciences

Advanced
November  2014, 34(11): 4419-4458. doi: 10.3934/dcds.2014.34.4419

Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids

Hyeong-Ohk Bae 1, , Young-Pil Choi 2, , Seung-Yeal Ha 3, and  Moon-Jin Kang 4,

1. 

Department of Financial Engineering, Ajou University, Suwon 443-749, South Korea
2. 

Department of Mathematics, Imperial College London, London SW7 2AZ
3. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea
4. 

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, United States

Received  November 2013 Revised  December 2013 Published  May 2014

We propose a coupled system for the interaction between Cucker-Smale flocking particles and viscous compressible fluids, and present a global existence theory and time-asymptotic behavior for the proposed model in the spatial periodic domain $\mathbb{T}^3$. Our model consists of the kinetic Cucker-Smale model for flocking particles and the isentropic compressible Navier-Stokes equations for fluids, and these two models are coupled through a drag force, which is responsible for the asymptotic alignment between particles and fluid. For the asymptotic flocking behavior, we explicitly construct a Lyapunov functional measuring the deviation from the asymptotic flocking states. For a large viscosity and small initial data, we show that the velocities of Cucker-Smale particles and fluids are asymptotically aligned to the common velocity.
Keywords: strong solution, asymptotic flocking behavior., kinetic Cucker-Smale model, compressible Navier-Stokes equations, Cucker-Smale flocking particle.
Mathematics Subject Classification: 35Q30, 35Q70, 76N10, 35Q8.
Citation: 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, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419
References:
[1]

Nonlinearity, 25 (2012), 1155-1177. doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[2]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system,, submitted., ().   Google Scholar

[3]

J. Hyperbolic Differ. Equ., 3 (2006), 1-26. doi: 10.1142/S0219891606000707.  Google Scholar

[4]

Trudy Sem. S. L. Sobolev, 80 (1980), 5-40.  Google Scholar

[5]

Hokkaido Math. J., 19 (1990), 67-87. doi: 10.14492/hokmj/1381517172.  Google Scholar

[6]

Differential and Integral Equations, 22 (2009), 1247-1271.  Google Scholar

[7]

SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.  Google Scholar

[8]

J. Differential Equation, 251 (2011), 2431-2465. doi: 10.1016/j.jde.2011.07.016.  Google Scholar

[9]

Adv. Differential Equations, 12 (2007), 893-960.  Google Scholar

[10]

J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[11]

J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[12]

IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[14]

Comm. Partial Differential Equations, 22 (1997), 977-1008. doi: 10.1080/03605309708821291.  Google Scholar

[15]

J. Math. Anal. Appl., 386 (2012), 939-947. doi: 10.1016/j.jmaa.2011.08.055.  Google Scholar

[16]

J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.  Google Scholar

[17]

SpringerVerlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[18]

Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508.  Google Scholar

[19]

Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.  Google Scholar

[20]

Comm. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[21]

Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.  Google Scholar

[22]

Japan J. Indust. Appl. Math., 15 (1998), 51-74. doi: 10.1007/BF03167396.  Google Scholar

[23]

J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[24]

Kodai math. Sem. Rep., 23 (1971), 60-120. doi: 10.2996/kmj/1138846265.  Google Scholar

[25]

Arch. Rational Mech. Anal., 165 (2002), 89-159. doi: 10.1007/s00205-002-0221-x.  Google Scholar

[26]

Arch. Rational Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6.  Google Scholar

[27]

J. Differential Equations, 184 (2002), 587-619. doi: 10.1006/jdeq.2002.4158.  Google Scholar

[28]

Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.  Google Scholar

[29]

Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[30]

J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[31]

Commun. Math. Phys., 89 (1983), 445-464.  Google Scholar

[32]

Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.  Google Scholar

[33]

Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[34]

J. Math. Soc. Jpn., 55 (2003), 797-826. doi: 10.2969/jmsj/1191419003.  Google Scholar

[35]

J. Hyperbolic Differ. Equ., 3 (2006), 561-574. doi: 10.1142/S0219891606000902.  Google Scholar

[36]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647.  Google Scholar

[37]

Commun. Math. Phys., 103 (1986), 259-296. doi: 10.1007/BF01206939.  Google Scholar

[38]

Addison-Wesley series in engineering scinece, 1965. Google Scholar

show all references

References:
[1]

Nonlinearity, 25 (2012), 1155-1177. doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[2]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system,, submitted., ().   Google Scholar

[3]

J. Hyperbolic Differ. Equ., 3 (2006), 1-26. doi: 10.1142/S0219891606000707.  Google Scholar

[4]

Trudy Sem. S. L. Sobolev, 80 (1980), 5-40.  Google Scholar

[5]

Hokkaido Math. J., 19 (1990), 67-87. doi: 10.14492/hokmj/1381517172.  Google Scholar

[6]

Differential and Integral Equations, 22 (2009), 1247-1271.  Google Scholar

[7]

SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.  Google Scholar

[8]

J. Differential Equation, 251 (2011), 2431-2465. doi: 10.1016/j.jde.2011.07.016.  Google Scholar

[9]

Adv. Differential Equations, 12 (2007), 893-960.  Google Scholar

[10]

J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[11]

J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[12]

IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[14]

Comm. Partial Differential Equations, 22 (1997), 977-1008. doi: 10.1080/03605309708821291.  Google Scholar

[15]

J. Math. Anal. Appl., 386 (2012), 939-947. doi: 10.1016/j.jmaa.2011.08.055.  Google Scholar

[16]

J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.  Google Scholar

[17]

SpringerVerlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[18]

Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508.  Google Scholar

[19]

Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.  Google Scholar

[20]

Comm. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[21]

Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.  Google Scholar

[22]

Japan J. Indust. Appl. Math., 15 (1998), 51-74. doi: 10.1007/BF03167396.  Google Scholar

[23]

J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[24]

Kodai math. Sem. Rep., 23 (1971), 60-120. doi: 10.2996/kmj/1138846265.  Google Scholar

[25]

Arch. Rational Mech. Anal., 165 (2002), 89-159. doi: 10.1007/s00205-002-0221-x.  Google Scholar

[26]

Arch. Rational Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6.  Google Scholar

[27]

J. Differential Equations, 184 (2002), 587-619. doi: 10.1006/jdeq.2002.4158.  Google Scholar

[28]

Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.  Google Scholar

[29]

Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[30]

J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[31]

Commun. Math. Phys., 89 (1983), 445-464.  Google Scholar

[32]

Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.  Google Scholar

[33]

Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[34]

J. Math. Soc. Jpn., 55 (2003), 797-826. doi: 10.2969/jmsj/1191419003.  Google Scholar

[35]

J. Hyperbolic Differ. Equ., 3 (2006), 561-574. doi: 10.1142/S0219891606000902.  Google Scholar

[36]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647.  Google Scholar

[37]

Commun. Math. Phys., 103 (1986), 259-296. doi: 10.1007/BF01206939.  Google Scholar

[38]

Addison-Wesley series in engineering scinece, 1965. Google Scholar

[1]

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
[2]

Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253
[3]

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
[4]

Hyunjin Ahn, Seung-Yeal Ha, Woojoo Shim. Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds. Kinetic & Related Models, 2021, 14 (2) : 323-351. doi: 10.3934/krm.2021007
[5]

Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246
[6]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[7]

Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005
[8]

Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
[9]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
[10]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[11]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[12]

Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021019
[13]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408
[14]

Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L∞ tensor coefficient under relaxed ellipticity condition. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021042
[15]

Francis Hounkpe, Gregory Seregin. An approximation of forward self-similar solutions to the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021059
[16]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[17]

Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009
[18]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091
[19]

Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021103
[20]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences