# American Institute of Mathematical Sciences

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 - A, 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/PL00000976. Google Scholar [17] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I,, SpringerVerlag, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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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##### 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/PL00000976. Google Scholar [17] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations I,, SpringerVerlag, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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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