doi: 10.3934/dcdsb.2021231
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Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

2. 

School of Mathematics and Physics, Wuhan Institute of Technology, Wuhan 430072, China

* Corresponding author: Hao Wang

Received  April 2021 Revised  July 2021 Early access September 2021

In this paper, we obtain the uniform estimates with respect to the Knudsen number $ \varepsilon $ for the fluctuations $ g^{\pm}_{\varepsilon} $ to the two-species Vlasov-Poisson-Boltzmann (in briefly, VPB) system. Then, we prove the existence of the global-in-time classical solutions for two-species VPB with all $ \varepsilon \in (0,1] $ on the torus under small initial data and rigorously derive the convergence to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system as $ \varepsilon $ go to 0.

Citation: Zhendong Fang, Hao Wang. Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021231
References:
[1]

D. Arsénio, From Boltzmann's Equation to the incompressible Navier-Stokes-Fourier system with long-range interactions, Arch. Ration. Mech. Anal., 206 (2012), 367-488.  doi: 10.1007/s00205-012-0557-9.  Google Scholar

[2]

D. Arsénio and L. Saint-Raymond, From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics, Vol. 1. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2019. doi: 10.4171/193.  Google Scholar

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C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I: Formal derivation, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations II: Convergence proof for the Boltzmann equation, Commun. Pure and Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[5]

C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257.  doi: 10.1142/S0218202591000137.  Google Scholar

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F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

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M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022.  Google Scholar

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M. Briant, S. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85–116. doi: 10.1142/S021953051850015X.  Google Scholar

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R. E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 651-666.  doi: 10.1002/cpa.3160330506.  Google Scholar

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A. De MasiR. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math., 42 (1989), 1189-1214.  doi: 10.1002/cpa.3160420810.  Google Scholar

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R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[12]

R. DuanT. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386.  doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[13]

R. DuanT. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23 (2013), 979-1028.  doi: 10.1142/S0218202513500012.  Google Scholar

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F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5.  Google Scholar

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M. M. Guo, N. Jiang and Y.-L. Luo, From Vlasov-Poisson-Boltzmann system to incompressible Navier-Stokes-Fourier-Poisson system: Convergence for classical solutions, arXiv: 2006.16514. Google Scholar

[16]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[17]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.  doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[18]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121.  Google Scholar

[19]

Y. GuoJ. Jang and N. Jiang, Local Hilbert expansion for the Boltzmann equation, Kinet. Relat. Models, 2 (2009), 205-214.  doi: 10.3934/krm.2009.2.205.  Google Scholar

[20]

Y. GuoJ. Jang and N. Jiang, Acoustic limit for the Boltzmann equation in optimal scaling, Comm. Pure Appl. Math., 63 (2010), 337-361.  doi: 10.1002/cpa.20308.  Google Scholar

[21]

J. Jang and N. Jiang, Acoustic limit of the Boltzmann equation: Classical solutions, Discrete Contin. Dyn. Syst., 25 (2009), 869-882.  doi: 10.3934/dcds.2009.25.869.  Google Scholar

[22]

N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I, Comm. Pure Appl. Math., 70 (2017), 90-171.  doi: 10.1002/cpa.21631.  Google Scholar

[23]

N. JiangC.-J. Xu and H. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, Indiana Univ. Math. J., 67 (2018), 1817-1855.  doi: 10.1512/iumj.2018.67.5940.  Google Scholar

[24]

N. Jiang and X. Zhang, Sensitivity analysis and incompressible Navier-Stokes-Poisson limit of Vlasov-Poisson-Boltzmann equations with uncertainty, arXiv: 2007.00879. Google Scholar

[25]

C. D. Levermore and W. Sun, Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels, Kinet. Relat. Models, 3 (2010), 335-351.  doi: 10.3934/krm.2010.3.335.  Google Scholar

[26]

H.-L. LiT. Yang and M. Zhong, Diffusion limit of the Vlasov-Poisson-Boltzmann system, Kinet. Relat. Models, 14 (2021), 211-255.  doi: 10.3934/krm.2021003.  Google Scholar

[27]

P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), 391–427,429–461. doi: 10.1215/kjm/1250519017.  Google Scholar

[28]

N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math., 56 (2003), 1263-1293.  doi: 10.1002/cpa.10095.  Google Scholar

[29]

T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148.  doi: 10.1007/BF01609490.  Google Scholar

[30]

L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equations, volume 1971 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-92847-8.  Google Scholar

[31]

Q. XiaoL. Xiong and H. Zhao, The Vlasov-Posson-Boltzmann system without angular cutoff for hard potential, Sci. China Math., 57 (2014), 515-540.  doi: 10.1007/s11425-013-4712-z.  Google Scholar

[32]

Q. XiaoL. Xiong and H. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, J. Funct. Anal., 272 (2017), 166-226.  doi: 10.1016/j.jfa.2016.09.017.  Google Scholar

show all references

References:
[1]

D. Arsénio, From Boltzmann's Equation to the incompressible Navier-Stokes-Fourier system with long-range interactions, Arch. Ration. Mech. Anal., 206 (2012), 367-488.  doi: 10.1007/s00205-012-0557-9.  Google Scholar

[2]

D. Arsénio and L. Saint-Raymond, From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics, Vol. 1. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2019. doi: 10.4171/193.  Google Scholar

[3]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I: Formal derivation, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[4]

C. BardosF. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations II: Convergence proof for the Boltzmann equation, Commun. Pure and Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[5]

C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257.  doi: 10.1142/S0218202591000137.  Google Scholar

[6]

F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[7]

M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022.  Google Scholar

[8]

M. Briant, S. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85–116. doi: 10.1142/S021953051850015X.  Google Scholar

[9]

R. E. Caflisch, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 651-666.  doi: 10.1002/cpa.3160330506.  Google Scholar

[10]

A. De MasiR. Esposito and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math., 42 (1989), 1189-1214.  doi: 10.1002/cpa.3160420810.  Google Scholar

[11]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[12]

R. DuanT. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386.  doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[13]

R. DuanT. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23 (2013), 979-1028.  doi: 10.1142/S0218202513500012.  Google Scholar

[14]

F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math., 155 (2004), 81-161.  doi: 10.1007/s00222-003-0316-5.  Google Scholar

[15]

M. M. Guo, N. Jiang and Y.-L. Luo, From Vlasov-Poisson-Boltzmann system to incompressible Navier-Stokes-Fourier-Poisson system: Convergence for classical solutions, arXiv: 2006.16514. Google Scholar

[16]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[17]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.  doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[18]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121.  Google Scholar

[19]

Y. GuoJ. Jang and N. Jiang, Local Hilbert expansion for the Boltzmann equation, Kinet. Relat. Models, 2 (2009), 205-214.  doi: 10.3934/krm.2009.2.205.  Google Scholar

[20]

Y. GuoJ. Jang and N. Jiang, Acoustic limit for the Boltzmann equation in optimal scaling, Comm. Pure Appl. Math., 63 (2010), 337-361.  doi: 10.1002/cpa.20308.  Google Scholar

[21]

J. Jang and N. Jiang, Acoustic limit of the Boltzmann equation: Classical solutions, Discrete Contin. Dyn. Syst., 25 (2009), 869-882.  doi: 10.3934/dcds.2009.25.869.  Google Scholar

[22]

N. Jiang and N. Masmoudi, Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I, Comm. Pure Appl. Math., 70 (2017), 90-171.  doi: 10.1002/cpa.21631.  Google Scholar

[23]

N. JiangC.-J. Xu and H. Zhao, Incompressible Navier-Stokes-Fourier limit from the Boltzmann equation: Classical solutions, Indiana Univ. Math. J., 67 (2018), 1817-1855.  doi: 10.1512/iumj.2018.67.5940.  Google Scholar

[24]

N. Jiang and X. Zhang, Sensitivity analysis and incompressible Navier-Stokes-Poisson limit of Vlasov-Poisson-Boltzmann equations with uncertainty, arXiv: 2007.00879. Google Scholar

[25]

C. D. Levermore and W. Sun, Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels, Kinet. Relat. Models, 3 (2010), 335-351.  doi: 10.3934/krm.2010.3.335.  Google Scholar

[26]

H.-L. LiT. Yang and M. Zhong, Diffusion limit of the Vlasov-Poisson-Boltzmann system, Kinet. Relat. Models, 14 (2021), 211-255.  doi: 10.3934/krm.2021003.  Google Scholar

[27]

P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), 391–427,429–461. doi: 10.1215/kjm/1250519017.  Google Scholar

[28]

N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math., 56 (2003), 1263-1293.  doi: 10.1002/cpa.10095.  Google Scholar

[29]

T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Comm. Math. Phys., 61 (1978), 119-148.  doi: 10.1007/BF01609490.  Google Scholar

[30]

L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equations, volume 1971 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-92847-8.  Google Scholar

[31]

Q. XiaoL. Xiong and H. Zhao, The Vlasov-Posson-Boltzmann system without angular cutoff for hard potential, Sci. China Math., 57 (2014), 515-540.  doi: 10.1007/s11425-013-4712-z.  Google Scholar

[32]

Q. XiaoL. Xiong and H. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, J. Funct. Anal., 272 (2017), 166-226.  doi: 10.1016/j.jfa.2016.09.017.  Google Scholar

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