doi: 10.3934/krm.2022016
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On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum

1. 

Department of Mathematics, Yonsei University, Seoul, 03722, Republic of Korea

2. 

Research Institute of Basic Sciences, Seoul National University, Seoul, 08826, Republic of Korea

*Corresponding author: Jinwook Jung

Received  January 2022 Early access May 2022

We analyze the Vlasov equation coupled with the compressible Navier–Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier–Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991–1021] completely rigorous.

Citation: Young-Pil Choi, Jinwook Jung. On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum. Kinetic and Related Models, doi: 10.3934/krm.2022016
References:
[1]

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

[2]

L. BoudinC. Grandmont and A. Moussa, Global existence of solutions to the incompressible Navier–Stokes–Vlasov equations in a time-dependent domain, J. Differential Equations, 262 (2017), 1317-1340.  doi: 10.1016/j.jde.2016.10.012.

[3]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.

[4]

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

[5]

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

[6]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Comm. Partial Differential Equations, 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.

[7]

M. ChaeK. Kang and J. Lee, Global existence of weak and classical solutions for the Navier–Stokes–Vlasov–Fokker–Planck equations, J. Differential Equations, 251 (2011), 2431-2465.  doi: 10.1016/j.jde.2011.07.016.

[8]

M. ChaeK. Kang and J. Lee, Global classical solutions for a compressible fluid-particle interaction model, J. Hyperbolic Diff. Eqns., 10 (2013), 537-562.  doi: 10.1142/S0219891613500197.

[9]

Y.-P. Choi, Large-time behavior for the Vlasov/compressible Navier–Stokes equations, J. Math. Phys., 57 (2016), 071501, 13 pp. doi: 10.1063/1.4955026.

[10]

Y.-P. Choi, Finite-time blow-up phenomena of Vlasov/Navier–Stokes equations and related systems, J. Math. Pures Appl., 108 (2017), 991-1021.  doi: 10.1016/j.matpur.2017.05.019.

[11]

Y.-P. Choi and J. Jung, On the dynamics of charged particles in an incompressible flow: From kinetic-fluid to fluid-fluid models, Commun. Contemp. Math., 2022. doi: 10.1142/S0219199722500122.

[12]

Y.-P. Choi and J. Jung, Asymptotic analysis for a Vlasov–Fokker–Planck/compressible Navier-Stokes system in a bounded domain, Math. Models Methods Appl. Sci., 31 (2021), 2213-2295.  doi: 10.1142/S0218202521500482.

[13]

Y.-P. Choi and J. Jung, Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier–Stokes equations with a density-dependent viscosity, AIMS Ser. Appl. Math., Am. Inst. Math. Sci. (AIMS), Hyperbolic problems: Theory, numerics, applications, Springfield, MO, 10 (2020), 145–163.

[14]

Y.-P. Choi and B. Kwon, Global well-posedness and large-time behavior for the inhomogeneous Vlasov–Navier–Stokes equations, Nonlinearity, 28 (2015), 3309-3336.  doi: 10.1088/0951-7715/28/9/3309.

[15]

Y.-P. ChoiJ. Lee and S.-B. Yun, Strong solutions to the inhomogeneous Navier–Stokes–BGK system, Nonlinear Anal. Real World Appl., 57 (2021), 103196.  doi: 10.1016/j.nonrwa.2020.103196.

[16]

Y.-P. Choi and S.-B. Yun, Global existence of weak solutions for Navier–Stokes–BGK system, Nonlinearity, 33 (2020), 1925-1955.  doi: 10.1088/1361-6544/ab6c38.

[17]

L. Desvillettes, Some aspects of the modelling at different scales of multiphase flows, Comput. Methods Appl. Mech. Eng., 199 (2010), 1265-1267.  doi: 10.1016/j.cma.2009.08.008.

[18]

T. GoudonP.-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. GoudonP.-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]

D. Han-Kwan, Large time behavior of small data solutions to the Vlasov–Navier–Stokes system on the whole space, Prob. Math. Phys..

[21]

D. Han-Kwan and D. Michel, On hydrodynamic limits of the Vlasov–Navier–Stokes system, Mem. Amer. Math. Soc., to appear.

[22]

D. Han-KwanA. Moussa and I. Moyano, Large time behavior of the Vlasov–Navier–Stokes system on the torus, Arch. Ration. Mech. Anal., 236 (2020), 1273-1323.  doi: 10.1007/s00205-020-01491-w.

[23]

F. LiY. Mu and D. 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.

[24]

Y. LiR. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational Mech. Anal., 234 (2019), 1281-1334.  doi: 10.1007/s00205-019-01412-6.

[25]

Y. LiR. Pan and S. Zhu, On classical solutions to 2D Shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.

[26]

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.

[27]

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

[28]

P. O'Rourke, Collective Drop Effects on Vaporising Liquid Sprays, Ph. D. Thesis, Princeton University, Princeton, NJ, 1981.

[29]

F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555.  doi: 10.1063/1.1724379.

[30]

Z. Xin, Blow up of smooth solutions to the compressible Navier–Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[31]

Z. Xin and W. Yan, On blow up of classical solutions to the compressible Navier–Stokes equations, Comm. Math. Phys., 321 (2013), 529-541.  doi: 10.1007/s00220-012-1610-0.

[32]

L. Yao and C. Yu, Existence of global weak solutions for the Navier–Stokes–Vlasov–Boltzmann equations, J. Differential Equations, 265 (2018), 5575-5603.  doi: 10.1016/j.jde.2018.07.001.

[33]

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]

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

[2]

L. BoudinC. Grandmont and A. Moussa, Global existence of solutions to the incompressible Navier–Stokes–Vlasov equations in a time-dependent domain, J. Differential Equations, 262 (2017), 1317-1340.  doi: 10.1016/j.jde.2016.10.012.

[3]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.

[4]

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

[5]

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

[6]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Comm. Partial Differential Equations, 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.

[7]

M. ChaeK. Kang and J. Lee, Global existence of weak and classical solutions for the Navier–Stokes–Vlasov–Fokker–Planck equations, J. Differential Equations, 251 (2011), 2431-2465.  doi: 10.1016/j.jde.2011.07.016.

[8]

M. ChaeK. Kang and J. Lee, Global classical solutions for a compressible fluid-particle interaction model, J. Hyperbolic Diff. Eqns., 10 (2013), 537-562.  doi: 10.1142/S0219891613500197.

[9]

Y.-P. Choi, Large-time behavior for the Vlasov/compressible Navier–Stokes equations, J. Math. Phys., 57 (2016), 071501, 13 pp. doi: 10.1063/1.4955026.

[10]

Y.-P. Choi, Finite-time blow-up phenomena of Vlasov/Navier–Stokes equations and related systems, J. Math. Pures Appl., 108 (2017), 991-1021.  doi: 10.1016/j.matpur.2017.05.019.

[11]

Y.-P. Choi and J. Jung, On the dynamics of charged particles in an incompressible flow: From kinetic-fluid to fluid-fluid models, Commun. Contemp. Math., 2022. doi: 10.1142/S0219199722500122.

[12]

Y.-P. Choi and J. Jung, Asymptotic analysis for a Vlasov–Fokker–Planck/compressible Navier-Stokes system in a bounded domain, Math. Models Methods Appl. Sci., 31 (2021), 2213-2295.  doi: 10.1142/S0218202521500482.

[13]

Y.-P. Choi and J. Jung, Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier–Stokes equations with a density-dependent viscosity, AIMS Ser. Appl. Math., Am. Inst. Math. Sci. (AIMS), Hyperbolic problems: Theory, numerics, applications, Springfield, MO, 10 (2020), 145–163.

[14]

Y.-P. Choi and B. Kwon, Global well-posedness and large-time behavior for the inhomogeneous Vlasov–Navier–Stokes equations, Nonlinearity, 28 (2015), 3309-3336.  doi: 10.1088/0951-7715/28/9/3309.

[15]

Y.-P. ChoiJ. Lee and S.-B. Yun, Strong solutions to the inhomogeneous Navier–Stokes–BGK system, Nonlinear Anal. Real World Appl., 57 (2021), 103196.  doi: 10.1016/j.nonrwa.2020.103196.

[16]

Y.-P. Choi and S.-B. Yun, Global existence of weak solutions for Navier–Stokes–BGK system, Nonlinearity, 33 (2020), 1925-1955.  doi: 10.1088/1361-6544/ab6c38.

[17]

L. Desvillettes, Some aspects of the modelling at different scales of multiphase flows, Comput. Methods Appl. Mech. Eng., 199 (2010), 1265-1267.  doi: 10.1016/j.cma.2009.08.008.

[18]

T. GoudonP.-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. GoudonP.-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]

D. Han-Kwan, Large time behavior of small data solutions to the Vlasov–Navier–Stokes system on the whole space, Prob. Math. Phys..

[21]

D. Han-Kwan and D. Michel, On hydrodynamic limits of the Vlasov–Navier–Stokes system, Mem. Amer. Math. Soc., to appear.

[22]

D. Han-KwanA. Moussa and I. Moyano, Large time behavior of the Vlasov–Navier–Stokes system on the torus, Arch. Ration. Mech. Anal., 236 (2020), 1273-1323.  doi: 10.1007/s00205-020-01491-w.

[23]

F. LiY. Mu and D. 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.

[24]

Y. LiR. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational Mech. Anal., 234 (2019), 1281-1334.  doi: 10.1007/s00205-019-01412-6.

[25]

Y. LiR. Pan and S. Zhu, On classical solutions to 2D Shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.

[26]

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.

[27]

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

[28]

P. O'Rourke, Collective Drop Effects on Vaporising Liquid Sprays, Ph. D. Thesis, Princeton University, Princeton, NJ, 1981.

[29]

F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555.  doi: 10.1063/1.1724379.

[30]

Z. Xin, Blow up of smooth solutions to the compressible Navier–Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[31]

Z. Xin and W. Yan, On blow up of classical solutions to the compressible Navier–Stokes equations, Comm. Math. Phys., 321 (2013), 529-541.  doi: 10.1007/s00220-012-1610-0.

[32]

L. Yao and C. Yu, Existence of global weak solutions for the Navier–Stokes–Vlasov–Boltzmann equations, J. Differential Equations, 265 (2018), 5575-5603.  doi: 10.1016/j.jde.2018.07.001.

[33]

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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