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

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

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.
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]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, 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]

C. Baranger and L. Desvillettes, Coupling Euler and Vlasov equation in the context of sprays: the local-in-time, classical solutions, J. Hyperbolic Differ. Equ., 3 (2006), 1-26. doi: 10.1142/S0219891606000707.  Google Scholar

[4]

M. E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad (in Russian), Trudy Sem. S. L. Sobolev, 80 (1980), 5-40.  Google Scholar

[5]

W. Borchers and H. Sohr, On the equation rot $v = g$ and div$u = f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87. doi: 10.14492/hokmj/1381517172.  Google Scholar

[6]

L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differential and Integral Equations, 22 (2009), 1247-1271.  Google Scholar

[7]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.  Google Scholar

[8]

M. Chae, K. Kang and J. Lee, Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations, J. Differential Equation, 251 (2011), 2431-2465. doi: 10.1016/j.jde.2011.07.016.  Google Scholar

[9]

Y. Cho, High regularity of solutions of compressible Navier-Stokes equations, Adv. Differential Equations, 12 (2007), 893-960.  Google Scholar

[10]

Y. Cho, H.-J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible vicous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[11]

H.-J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[12]

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

[13]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[14]

B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Comm. Partial Differential Equations, 22 (1997), 977-1008. doi: 10.1080/03605309708821291.  Google Scholar

[15]

D. Fang, R. Zi and T. Zhang, Decay estimates for isentropic compressible Navier-Stokes equations in bounded domain, J. Math. Anal. Appl., 386 (2012), 939-947. doi: 10.1016/j.jmaa.2011.08.055.  Google Scholar

[16]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.  Google Scholar

[17]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I, SpringerVerlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[18]

T. Goudon, P.-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.  Google Scholar

[19]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.  Google Scholar

[20]

S.-Y. Ha and J.-G. Liu, Short proof of Cucker-Smales flocking and the mean-field limit, Comm. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[21]

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

[22]

K. Hamdache, Global existence and large time behavior of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74. doi: 10.1007/BF03167396.  Google Scholar

[23]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[24]

N. Itaya, On the cauchy problem for the system of fundamental equations descrbing the movement of compressible viscous fluids, Kodai math. Sem. Rep., 23 (1971), 60-120. doi: 10.2996/kmj/1138846265.  Google Scholar

[25]

Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\mathbbR^3$, Arch. Rational Mech. Anal., 165 (2002), 89-159. doi: 10.1007/s00205-002-0221-x.  Google Scholar

[26]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Rational Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6.  Google Scholar

[27]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbbR^3$, J. Differential Equations, 184 (2002), 587-619. doi: 10.1006/jdeq.2002.4158.  Google Scholar

[28]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbbR^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.  Google Scholar

[29]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, 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]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[31]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.  Google Scholar

[32]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.  Google Scholar

[33]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[34]

Y. Shibata and K. Tanaka, On the steady compressible viscous fluid and its stability with respect to initial distrubance, J. Math. Soc. Jpn., 55 (2003), 797-826. doi: 10.2969/jmsj/1191419003.  Google Scholar

[35]

S. Ukai, T. Yang and H.-J. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574. doi: 10.1142/S0219891606000902.  Google Scholar

[36]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647.  Google Scholar

[37]

A. Valli and W. M. Zajaczkowski, Navier-Stokes equations for compressible fluids, global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys., 103 (1986), 259-296. doi: 10.1007/BF01206939.  Google Scholar

[38]

F. A. Williams, Combustion Theory, The Fundamental Theory of Chemically Reacting Flow Systems, Addison-Wesley series in engineering scinece, 1965. Google Scholar

show all references

References:
[1]

H.-O. Bae, Y.-P. Choi, S.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, 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]

C. Baranger and L. Desvillettes, Coupling Euler and Vlasov equation in the context of sprays: the local-in-time, classical solutions, J. Hyperbolic Differ. Equ., 3 (2006), 1-26. doi: 10.1142/S0219891606000707.  Google Scholar

[4]

M. E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad (in Russian), Trudy Sem. S. L. Sobolev, 80 (1980), 5-40.  Google Scholar

[5]

W. Borchers and H. Sohr, On the equation rot $v = g$ and div$u = f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87. doi: 10.14492/hokmj/1381517172.  Google Scholar

[6]

L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differential and Integral Equations, 22 (2009), 1247-1271.  Google Scholar

[7]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.  Google Scholar

[8]

M. Chae, K. Kang and J. Lee, Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations, J. Differential Equation, 251 (2011), 2431-2465. doi: 10.1016/j.jde.2011.07.016.  Google Scholar

[9]

Y. Cho, High regularity of solutions of compressible Navier-Stokes equations, Adv. Differential Equations, 12 (2007), 893-960.  Google Scholar

[10]

Y. Cho, H.-J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible vicous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[11]

H.-J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations, 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar

[12]

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

[13]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[14]

B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Comm. Partial Differential Equations, 22 (1997), 977-1008. doi: 10.1080/03605309708821291.  Google Scholar

[15]

D. Fang, R. Zi and T. Zhang, Decay estimates for isentropic compressible Navier-Stokes equations in bounded domain, J. Math. Anal. Appl., 386 (2012), 939-947. doi: 10.1016/j.jmaa.2011.08.055.  Google Scholar

[16]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.  Google Scholar

[17]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I, SpringerVerlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[18]

T. Goudon, P.-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.  Google Scholar

[19]

T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536. doi: 10.1512/iumj.2004.53.2509.  Google Scholar

[20]

S.-Y. Ha and J.-G. Liu, Short proof of Cucker-Smales flocking and the mean-field limit, Comm. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[21]

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

[22]

K. Hamdache, Global existence and large time behavior of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74. doi: 10.1007/BF03167396.  Google Scholar

[23]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.  Google Scholar

[24]

N. Itaya, On the cauchy problem for the system of fundamental equations descrbing the movement of compressible viscous fluids, Kodai math. Sem. Rep., 23 (1971), 60-120. doi: 10.2996/kmj/1138846265.  Google Scholar

[25]

Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\mathbbR^3$, Arch. Rational Mech. Anal., 165 (2002), 89-159. doi: 10.1007/s00205-002-0221-x.  Google Scholar

[26]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Rational Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6.  Google Scholar

[27]

T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in $\mathbbR^3$, J. Differential Equations, 184 (2002), 587-619. doi: 10.1006/jdeq.2002.4158.  Google Scholar

[28]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbbR^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.  Google Scholar

[29]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, 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]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[31]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.  Google Scholar

[32]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063. doi: 10.1142/S0218202507002194.  Google Scholar

[33]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[34]

Y. Shibata and K. Tanaka, On the steady compressible viscous fluid and its stability with respect to initial distrubance, J. Math. Soc. Jpn., 55 (2003), 797-826. doi: 10.2969/jmsj/1191419003.  Google Scholar

[35]

S. Ukai, T. Yang and H.-J. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574. doi: 10.1142/S0219891606000902.  Google Scholar

[36]

A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), 607-647.  Google Scholar

[37]

A. Valli and W. M. Zajaczkowski, Navier-Stokes equations for compressible fluids, global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys., 103 (1986), 259-296. doi: 10.1007/BF01206939.  Google Scholar

[38]

F. A. Williams, Combustion Theory, The Fundamental Theory of Chemically Reacting Flow Systems, Addison-Wesley series in engineering scinece, 1965. Google Scholar

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