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April  2021, 14(2): 211-255. doi: 10.3934/krm.2021003

Diffusion limit of the Vlasov-Poisson-Boltzmann system

Hai-Liang Li 1, , Tong Yang 2, and  Mingying Zhong 3,,

1. 

School of Mathematical Sciences, Capital Normal University, China
2. 

Department of Mathematics, City University of Hong Kong, China, School of Mathematics and Statistics, Chongqing University, China
3. 

College of Mathematics and Information Sciences, , Guangxi University, China

* Corresponding author: Mingying Zhong

Received  July 2020 Published  December 2020

Fund Project: The first author was supported partially by the National Science Fund for Distinguished Young Scholars No. 11225102, the National Natural Science Foundation of China Nos. 11931010, 11871047 and 11671384, and the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds 007/20530290068 and 00719530050166. The second author was supported by the General Research Fund of Hong Kong, CityU 11302518, and the Fundamental Research Funds for the Central Universities No.2019CDJCYJ001. The third author is supported by the National Natural Science Foundation of China No. 11671100, the National Science Fund for Excellent Young Scholars No. 11922107, and Guangxi Natural Science Foundation Nos. 2018GXNSFAA138210 and 2020GXNSFFA238001

In the present paper, we study the diffusion limit of the classical solution to the unipolar Vlasov-Poisson-Boltzmann (VPB) system with initial data near a global Maxwellian. We prove the convergence and establish the convergence rate of the global strong solution to the unipolar VPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system based on the spectral analysis with precise estimation on the initial layer.

Keywords: Vlasov-Poisson-Boltzmann system, spectral analysis, diffusion limit, convergence rate.
Mathematics Subject Classification: 76P05, 82C40, 82D05.
Citation: Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
References:
[1]

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

[2]

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

[3]

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

[4]

Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains, Arch. Rational Mech. Anal., 233 (2019), 1027-1130.  doi: 10.1007/s00205-019-01374-9.  Google Scholar

[5]

R. J. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in ${\mathbb{R}}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.  doi: 10.1007/s00205-010-0318-6.  Google Scholar

[6]

R. J. Duan and T. Yang, Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41 (2010), 2353-2387.  doi: 10.1137/090745775.  Google Scholar

[7]

R. J. Duan, T. Yang and H. J. 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

[8]

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

[9]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[10]

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

[11]

Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys., 218 (2001), 293-313.  doi: 10.1007/s002200100391.  Google Scholar

[12]

Y. Guo and J. Jang, Global hilbert expansion for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 299 (2010), 469-501.  doi: 10.1007/s00220-010-1089-5.  Google Scholar

[13]

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

[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]

T. Kato, Perturbation Theory of Linear Operator, Springer, New York, 1966.  Google Scholar

[16]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, Preprint, arXiv: 1402.3633v1. Google Scholar

[17]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations, Indiana Univ. Math. J., 65 (2016), 665-725.  doi: 10.1512/iumj.2016.65.5730.  Google Scholar

[18]

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

[19]

P.-L. Lions, Compactness in Boltzmann's equation via fourier integral operators and applications. III, J. Math. Kyoto Univ., 34 (1994), 539-584.  doi: 10.1215/kjm/1250518932.  Google Scholar

[20]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[21]

A. De Masi, R. 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

[22]

S. Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 210, (2000), 447–466. doi: 10.1007/s002200050787.  Google Scholar

[23]

S. Nelson, On some solutions to the Klein-Gordon equations related to an integral of Sonine, Trans. A. M. S., 154 (1971), 227-237.  doi: 10.1090/S0002-9947-1971-0415049-9.  Google Scholar

[24]

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

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[26]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.  doi: 10.3792/pja/1195519027.  Google Scholar

[27]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, March, 2006. Google Scholar

[28]

Y. J. Wang, The Diffusive Limit of the Vlasov-Boltzmann System for Binary Fluids, SIAM J. Math. Anal., 43 (2011), 253-301.  doi: 10.1137/10079166X.  Google Scholar

[29]

Y. J. Wang, Decay of the two-species Vlasov-Poisson-Boltzmann system, J. Differential Equations, 254 (2013), 2304-2340.  doi: 10.1016/j.jde.2012.12.007.  Google Scholar

[30]

T. Yang, H. J. Yu and H. J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rational Mech. Anal., 182 (2006), 415-470.  doi: 10.1007/s00205-006-0009-5.  Google Scholar

[31]

T. Yang and H. J. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 268 (2006), 569-605.  doi: 10.1007/s00220-006-0103-4.  Google Scholar

[32]

T. Yang and H. J. Yu, Optimal convergence rates of classical solutions for Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 301 (2011), 319-355.  doi: 10.1007/s00220-010-1142-4.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains, Arch. Rational Mech. Anal., 233 (2019), 1027-1130.  doi: 10.1007/s00205-019-01374-9.  Google Scholar

[5]

R. J. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in ${\mathbb{R}}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.  doi: 10.1007/s00205-010-0318-6.  Google Scholar

[6]

R. J. Duan and T. Yang, Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41 (2010), 2353-2387.  doi: 10.1137/090745775.  Google Scholar

[7]

R. J. Duan, T. Yang and H. J. 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

[8]

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

[9]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. Math., 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[10]

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

[11]

Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys., 218 (2001), 293-313.  doi: 10.1007/s002200100391.  Google Scholar

[12]

Y. Guo and J. Jang, Global hilbert expansion for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 299 (2010), 469-501.  doi: 10.1007/s00220-010-1089-5.  Google Scholar

[13]

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

[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]

T. Kato, Perturbation Theory of Linear Operator, Springer, New York, 1966.  Google Scholar

[16]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, Preprint, arXiv: 1402.3633v1. Google Scholar

[17]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations, Indiana Univ. Math. J., 65 (2016), 665-725.  doi: 10.1512/iumj.2016.65.5730.  Google Scholar

[18]

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

[19]

P.-L. Lions, Compactness in Boltzmann's equation via fourier integral operators and applications. III, J. Math. Kyoto Univ., 34 (1994), 539-584.  doi: 10.1215/kjm/1250518932.  Google Scholar

[20]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[21]

A. De Masi, R. 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

[22]

S. Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 210, (2000), 447–466. doi: 10.1007/s002200050787.  Google Scholar

[23]

S. Nelson, On some solutions to the Klein-Gordon equations related to an integral of Sonine, Trans. A. M. S., 154 (1971), 227-237.  doi: 10.1090/S0002-9947-1971-0415049-9.  Google Scholar

[24]

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

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[26]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.  doi: 10.3792/pja/1195519027.  Google Scholar

[27]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation, Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, March, 2006. Google Scholar

[28]

Y. J. Wang, The Diffusive Limit of the Vlasov-Boltzmann System for Binary Fluids, SIAM J. Math. Anal., 43 (2011), 253-301.  doi: 10.1137/10079166X.  Google Scholar

[29]

Y. J. Wang, Decay of the two-species Vlasov-Poisson-Boltzmann system, J. Differential Equations, 254 (2013), 2304-2340.  doi: 10.1016/j.jde.2012.12.007.  Google Scholar

[30]

T. Yang, H. J. Yu and H. J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rational Mech. Anal., 182 (2006), 415-470.  doi: 10.1007/s00205-006-0009-5.  Google Scholar

[31]

T. Yang and H. J. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 268 (2006), 569-605.  doi: 10.1007/s00220-006-0103-4.  Google Scholar

[32]

T. Yang and H. J. Yu, Optimal convergence rates of classical solutions for Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 301 (2011), 319-355.  doi: 10.1007/s00220-010-1142-4.  Google Scholar

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