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Diffusion limit of the Vlasov-Poisson-Boltzmann system
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[10] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[11] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys., 218 (2001), 293-313.
doi: 10.1007/s002200100391. |
[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. |
[13] |
Y. Guo,
Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.
doi: 10.1002/cpa.20121. |
[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. |
[15] |
T. Kato, Perturbation Theory of Linear Operator, Springer, New York, 1966. |
[16] |
H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, Preprint, arXiv: 1402.3633v1. |
[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. |
[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. |
[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. |
[20] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[10] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[11] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys., 218 (2001), 293-313.
doi: 10.1007/s002200100391. |
[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. |
[13] |
Y. Guo,
Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.
doi: 10.1002/cpa.20121. |
[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. |
[15] |
T. Kato, Perturbation Theory of Linear Operator, Springer, New York, 1966. |
[16] |
H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, Preprint, arXiv: 1402.3633v1. |
[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. |
[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. |
[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. |
[20] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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