February  2020, 19(2): 835-882. doi: 10.3934/cpaa.2020039

A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

School of Mathematics, Hefei University of Technology, Hefei 230009, China

4. 

Wuhan Institute of Physics and Mathematicss, Chinese Academy of Science, Wuhan 430071, China

5. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author

Received  December 2018 Revised  June 2019 Published  October 2019

Fund Project: The work of S.-Y. Ha is supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03, the work of Qinghua Xiao was supported by grants from Youth Innovation Promotion Association and the National Natural Science Foundation of China #11871469, #11501556, and the work of Xiongtao Zhang was supported by grant the National Natural Science Foundation of China #11801194.

We present a two-dimensional coupled system for flocking particle-compressible fluid interactions, and study its global solvability for the proposed coupled system. For particle and fluid dynamics, we employ the kinetic Cucker-Smale-Fokker-Planck (CS-FP) model for flocking particle part, and the isentropic compressible Navier-Stokes (N-S) equations for the fluid part, respectively, and these separate systems are coupled through the drag force. For the global solvability of the coupled system, we present a sufficient framework for the global existence of classical solutions with large initial data which can contain vacuum using the weighted energy method. We extend an earlier global solvability result [20] in the one-dimensional setting to the two-dimensional setting.

Citation: Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039
References:
[1]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301. doi: 10.1063/1.3496895.  Google Scholar

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 24 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differential Equations, 257 (2014), 2225-2255.  doi: 10.1016/j.jde.2014.05.035.  Google Scholar

[4]

C. BarangerL. BoudinP.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47.   Google Scholar

[5]

S. BerresR. BurgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.  Google Scholar

[6]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[7]

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

[8]

L. Boudin, L. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657–669.  Google Scholar

[9]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259–275.  Google Scholar

[10]

J. A. CarrilloR.-J. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system, Kinetic and Related Models, 4 (2011), 227-258.  doi: 10.3934/krm.2011.4.227.  Google Scholar

[11]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and non-existence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[12]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[13]

Y.-P. Choi, S.-Y. Ha and Z.-C. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkh$\ddot{a}$user Basel, (2017), 299–331.  Google Scholar

[14]

R. CoifmanR. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.  doi: 10.2307/1970954.  Google Scholar

[15]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.2307/1998628.  Google Scholar

[16]

R.-J. Duan and S.-Q. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinet. Relat. Models, 6 (2013), 687-700.  doi: 10.3934/krm.2013.6.687.  Google Scholar

[17]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541–544.  Google Scholar

[18]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, 2004.  Google Scholar

[19]

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

[20]

S.-Y. HaB.-K. HuangQ.-H. Xiao and X.-T. Zhang, Global classical solutions to 1D coupled system of flocking particles and compressible fluids with large initial data, Math. Models Methods Appl. Sci., 28 (2018), 1-60.  doi: 10.1142/S021820251850001X.  Google Scholar

[21]

S.-Y. Ha, J. Jeong, S.-E. Noh, Q.-H. Xiao and X.-T. Zhang, Emergent of dynamic of infinity many Cucker-Smale particles in a random environment, J. Differential Equations, 262 (2017), 2554–2591. doi: 10.1016/j.jde.2016.11.017.  Google Scholar

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297–325.  Google Scholar

[23]

X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions tothe two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123–154. doi: 10.1016/j.matpur.2016.02.003.  Google Scholar

[24]

T. GoudonL. HeA. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202.  doi: 10.1137/090776755.  Google Scholar

[25]

D. Gilbarg and N. Tridinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.  Google Scholar

[26]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521.  doi: 10.1007/s00021-014-0171-8.  Google Scholar

[27]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.  Google Scholar

[28]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press, New York, 1996.  Google Scholar

[29]

Y. Mei, Global classical solutions to the 2D compressible MHD equations with large data and vacuum, J. Differential Equations, 258 (2015), 3304-3359.  doi: 10.1016/j.jde.2014.11.023.  Google Scholar

[30]

Y. Mei, Corrigendum to "Global classical solutions to the 2D compressible MHD equations with large data and vacuum", J. Differential Equations, 258 (2015), 3360-3362.  doi: 10.1016/j.jde.2015.02.001.  Google Scholar

[31]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.  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]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl., vol. 27 (2004) Oxford Univ. Press, Oxford.  Google Scholar

[34]

J. Simon, Compact sets in the space $L_p(0, t;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[35]

E. M. Stein, Harmonic Analysis, Princeton University Press, 1995.  Google Scholar

[36]

D.-H. Wang and C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations, J. Differ. Equations, 259 (2015), 3976-4008.  doi: 10.1016/j.jde.2015.05.016.  Google Scholar

[37]

F. A. Williams, Combustion Theory, Benjamin Cummings, 1985. Google Scholar

[38]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.  Google Scholar

[39]

C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.  Google Scholar

show all references

References:
[1]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301. doi: 10.1063/1.3496895.  Google Scholar

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 24 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differential Equations, 257 (2014), 2225-2255.  doi: 10.1016/j.jde.2014.05.035.  Google Scholar

[4]

C. BarangerL. BoudinP.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47.   Google Scholar

[5]

S. BerresR. BurgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.  Google Scholar

[6]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[7]

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

[8]

L. Boudin, L. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657–669.  Google Scholar

[9]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259–275.  Google Scholar

[10]

J. A. CarrilloR.-J. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system, Kinetic and Related Models, 4 (2011), 227-258.  doi: 10.3934/krm.2011.4.227.  Google Scholar

[11]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and non-existence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[12]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[13]

Y.-P. Choi, S.-Y. Ha and Z.-C. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkh$\ddot{a}$user Basel, (2017), 299–331.  Google Scholar

[14]

R. CoifmanR. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.  doi: 10.2307/1970954.  Google Scholar

[15]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.2307/1998628.  Google Scholar

[16]

R.-J. Duan and S.-Q. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinet. Relat. Models, 6 (2013), 687-700.  doi: 10.3934/krm.2013.6.687.  Google Scholar

[17]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541–544.  Google Scholar

[18]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, 2004.  Google Scholar

[19]

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

[20]

S.-Y. HaB.-K. HuangQ.-H. Xiao and X.-T. Zhang, Global classical solutions to 1D coupled system of flocking particles and compressible fluids with large initial data, Math. Models Methods Appl. Sci., 28 (2018), 1-60.  doi: 10.1142/S021820251850001X.  Google Scholar

[21]

S.-Y. Ha, J. Jeong, S.-E. Noh, Q.-H. Xiao and X.-T. Zhang, Emergent of dynamic of infinity many Cucker-Smale particles in a random environment, J. Differential Equations, 262 (2017), 2554–2591. doi: 10.1016/j.jde.2016.11.017.  Google Scholar

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297–325.  Google Scholar

[23]

X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions tothe two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123–154. doi: 10.1016/j.matpur.2016.02.003.  Google Scholar

[24]

T. GoudonL. HeA. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202.  doi: 10.1137/090776755.  Google Scholar

[25]

D. Gilbarg and N. Tridinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.  Google Scholar

[26]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521.  doi: 10.1007/s00021-014-0171-8.  Google Scholar

[27]

Q.-S. JiuY. Wang and Z.-P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differential Equations, 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.  Google Scholar

[28]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press, New York, 1996.  Google Scholar

[29]

Y. Mei, Global classical solutions to the 2D compressible MHD equations with large data and vacuum, J. Differential Equations, 258 (2015), 3304-3359.  doi: 10.1016/j.jde.2014.11.023.  Google Scholar

[30]

Y. Mei, Corrigendum to "Global classical solutions to the 2D compressible MHD equations with large data and vacuum", J. Differential Equations, 258 (2015), 3360-3362.  doi: 10.1016/j.jde.2015.02.001.  Google Scholar

[31]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.  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]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl., vol. 27 (2004) Oxford Univ. Press, Oxford.  Google Scholar

[34]

J. Simon, Compact sets in the space $L_p(0, t;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[35]

E. M. Stein, Harmonic Analysis, Princeton University Press, 1995.  Google Scholar

[36]

D.-H. Wang and C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations, J. Differ. Equations, 259 (2015), 3976-4008.  doi: 10.1016/j.jde.2015.05.016.  Google Scholar

[37]

F. A. Williams, Combustion Theory, Benjamin Cummings, 1985. Google Scholar

[38]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.  Google Scholar

[39]

C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.  Google Scholar

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