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Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise
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 |
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 [
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. |
[2] |
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, 24 (2012), 1155-1177.
doi: 10.1088/0951-7715/25/4/1155. |
[3] |
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, J. Differential Equations, 257 (2014), 2225-2255.
doi: 10.1016/j.jde.2014.05.035. |
[4] |
C. Baranger, L. Boudin, P.-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.
|
[5] |
S. Berres, R. Burger, K. 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. |
[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. |
[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. |
[8] |
L. Boudin, L. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657–669. |
[9] |
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259–275. |
[10] |
J. A. Carrillo, R.-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. |
[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. |
[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. |
[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. |
[14] |
R. Coifman, R. Rochberg and G. Weiss,
Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.
doi: 10.2307/1970954. |
[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. |
[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. |
[17] |
H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541–544. |
[18] |
E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, 2004. |
[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. |
[20] |
S.-Y. Ha, B.-K. Huang, Q.-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. |
[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. |
[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. |
[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. |
[24] |
T. Goudon, L. He, A. 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. |
[25] |
D. Gilbarg and N. Tridinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998. |
[26] |
Q.-S. Jiu, Y. 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. |
[27] |
Q.-S. Jiu, Y. 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. |
[28] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press, New York, 1996. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[35] |
E. M. Stein, Harmonic Analysis, Princeton University Press, 1995. |
[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. |
[37] | |
[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. |
[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. |
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. |
[2] |
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, 24 (2012), 1155-1177.
doi: 10.1088/0951-7715/25/4/1155. |
[3] |
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, J. Differential Equations, 257 (2014), 2225-2255.
doi: 10.1016/j.jde.2014.05.035. |
[4] |
C. Baranger, L. Boudin, P.-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.
|
[5] |
S. Berres, R. Burger, K. 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. |
[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. |
[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. |
[8] |
L. Boudin, L. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657–669. |
[9] |
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259–275. |
[10] |
J. A. Carrillo, R.-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. |
[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. |
[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. |
[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. |
[14] |
R. Coifman, R. Rochberg and G. Weiss,
Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.
doi: 10.2307/1970954. |
[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. |
[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. |
[17] |
H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541–544. |
[18] |
E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, 2004. |
[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. |
[20] |
S.-Y. Ha, B.-K. Huang, Q.-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. |
[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. |
[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. |
[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. |
[24] |
T. Goudon, L. He, A. 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. |
[25] |
D. Gilbarg and N. Tridinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998. |
[26] |
Q.-S. Jiu, Y. 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. |
[27] |
Q.-S. Jiu, Y. 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. |
[28] |
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press, New York, 1996. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[35] |
E. M. Stein, Harmonic Analysis, Princeton University Press, 1995. |
[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. |
[37] | |
[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. |
[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. |
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