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May  2020, 19(5): 2737-2750. doi: 10.3934/cpaa.2020119

Asymptotic analysis for 1D compressible Navier-Stokes-Vlasov equations

1. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

2. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

3. 

School of Mathematics, Northwest University, Xi'an 710127, China

* Corresponding author

Received  April 2019 Revised  October 2019 Published  March 2020

Fund Project: H. Cui is supported by the National Natural Science Foundation of China (Grant No. 11971183, 11601164 and 11571380), the Natural Science Foundation of Fujian Province of China (Grant No. 2017J05007). W. Wang is supported by the National Natural Science Foundation of China (Grant No. 11871341, 11671150 and 11571231). L. Yao is supported by the National Natural Science Foundation of China (Grant No. 11571280 and 11931013), Natural Science Basic Research Program of Shaanxi (Program No. 2019JC-26) and FANEDD #201315

In this paper, we study the asymptotic analysis of 1D compressible Navier-Stokes-Vlasov equations. By taking advantage of the one space dimension, we obtain the hydrodynamic limit for compressible Navier-Stokes-Vlasov equations with the pressure $ P(\rho) = A\rho^{\gamma} $ $ (\gamma>1) $. The proof relies on weak convergence method.

Citation: Haibo Cui, Wenjun Wang, Lei Yao. Asymptotic analysis for 1D compressible Navier-Stokes-Vlasov equations. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2737-2750. doi: 10.3934/cpaa.2020119
References:
[1]

S. BenjellounL. Desvillettes and A. Moussa, Existence theory for the kinetic-fluid coupling when small droplets are treated as part of the fluid, J. Hyperbolic Differ. Equ., 11 (2014), 109-133.  doi: 10.1142/S0219891614500027.  Google Scholar

[2]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differ. Integral Equ., 22 (2009), 1247-1271.   Google Scholar

[3]

R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects, SIAM J. Appl. Math., 43 (1983), 885-906.  doi: 10.1137/0143057.  Google Scholar

[4]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.  Google Scholar

[5]

J. A. CarrilloY. P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 33 (2016), 273-307.  doi: 10.1016/j.anihpc.2014.10.002.  Google Scholar

[6]

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

[7]

R. M. Chen, Y. F. Su and L. Yao, Hydrodynamic limit for 1D compressible Navier-Stokes-Vlasov equations, Preprint, 2018. doi: 10.1063/1.4955026.  Google Scholar

[8]

E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98.   Google Scholar

[9]

E. FeireislA. 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

[10]

T. Goudon, Asymptotic problems for a kinetic model of two-phase flow, Proc. R. Soc. Edinb. Sect. A Math., 131 (2001), 1371-1384.  doi: 10.1017/S030821050000144X.  Google Scholar

[11]

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

[12]

T. GoudonP. E. Jabin and A. F. 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

[13]

T. GoudonP. E. Jabin and A. F. 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

[14]

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

[15]

M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7985-8.  Google Scholar

[16]

M. J. Kang and A. F. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.  Google Scholar

[17]

N. Leger and A. F. Vasseur, Study of a generalized fragmentation model for sprays, J. Hyperbolic Differ. Equ., 6 (2009), 185-206.  doi: 10.1142/S0219891609001770.  Google Scholar

[18]

F. C. LiY. M. Mu and D. H. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026.  doi: 10.1137/15M1053049.  Google Scholar

[19]

F. H. LinC. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866.  doi: 10.1002/cpa.20159.  Google Scholar

[20] P.L. Lions, Mathematical Topics in Fluid Mechanics, Compressible Models, Vol. II, Clarendon Press, Oxford, 1998.   Google Scholar
[21]

A. Mellet and A. F. 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

[22]

A. Mellet and A. F. 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

[23]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar

[24]

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

[25]

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. BenjellounL. Desvillettes and A. Moussa, Existence theory for the kinetic-fluid coupling when small droplets are treated as part of the fluid, J. Hyperbolic Differ. Equ., 11 (2014), 109-133.  doi: 10.1142/S0219891614500027.  Google Scholar

[2]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differ. Integral Equ., 22 (2009), 1247-1271.   Google Scholar

[3]

R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects, SIAM J. Appl. Math., 43 (1983), 885-906.  doi: 10.1137/0143057.  Google Scholar

[4]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.  Google Scholar

[5]

J. A. CarrilloY. P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 33 (2016), 273-307.  doi: 10.1016/j.anihpc.2014.10.002.  Google Scholar

[6]

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

[7]

R. M. Chen, Y. F. Su and L. Yao, Hydrodynamic limit for 1D compressible Navier-Stokes-Vlasov equations, Preprint, 2018. doi: 10.1063/1.4955026.  Google Scholar

[8]

E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin., 42 (2001), 83-98.   Google Scholar

[9]

E. FeireislA. 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

[10]

T. Goudon, Asymptotic problems for a kinetic model of two-phase flow, Proc. R. Soc. Edinb. Sect. A Math., 131 (2001), 1371-1384.  doi: 10.1017/S030821050000144X.  Google Scholar

[11]

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

[12]

T. GoudonP. E. Jabin and A. F. 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

[13]

T. GoudonP. E. Jabin and A. F. 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

[14]

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

[15]

M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7985-8.  Google Scholar

[16]

M. J. Kang and A. F. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.  Google Scholar

[17]

N. Leger and A. F. Vasseur, Study of a generalized fragmentation model for sprays, J. Hyperbolic Differ. Equ., 6 (2009), 185-206.  doi: 10.1142/S0219891609001770.  Google Scholar

[18]

F. C. LiY. M. Mu and D. H. Wang, Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior, SIAM J. Math. Anal., 49 (2017), 984-1026.  doi: 10.1137/15M1053049.  Google Scholar

[19]

F. H. LinC. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math., 60 (2007), 838-866.  doi: 10.1002/cpa.20159.  Google Scholar

[20] P.L. Lions, Mathematical Topics in Fluid Mechanics, Compressible Models, Vol. II, Clarendon Press, Oxford, 1998.   Google Scholar
[21]

A. Mellet and A. F. 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

[22]

A. Mellet and A. F. 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

[23]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar

[24]

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

[25]

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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