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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[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 $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[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 $\mathbb{R}^3$, Comm. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

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