American Institute of Mathematical Sciences

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March  2013, 6(1): 159-204. doi: 10.3934/krm.2013.6.159

Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials

Renjun Duan 1, , Shuangqian Liu 2, , Tong Yang 3, and  Huijiang Zhao 4,

1. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2. 

Department of Mathematics, Jinan Unviersity, Guangdong, China
3. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
4. 

School of Mathematics and Statistics, Wuhan University, China

Received  August 2012 Revised  October 2012 Published  December 2012

Although there recently have been extensive studies on the perturbation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with suitable regularity and integrability, we establish the large time stability of solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system with physical angular non-cutoff intermolecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal energy $X(t)$-norm to obtain its uniform bound. The result also extends the case of the hard sphere model considered by Guo [Invent. Math. 153(3): 593--630 (2003)] to the general collision potentials.
Keywords: non-cutoff potentials, time-velocity weight., energy method, The Vlasov-Maxwell-Boltzmann system.
Mathematics Subject Classification: Primary: 35Q20, 76P05; Secondary: 35B35, 35B4.
Citation: Renjun Duan, Shuangqian Liu, Tong Yang, Huijiang Zhao. Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. Kinetic & Related Models, 2013, 6 (1) : 159-204. doi: 10.3934/krm.2013.6.159
References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.  Google Scholar

[2]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123. doi: 10.1007/s00205-010-0290-1.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 263 (2012), 915-1010. doi: 10.1016/j.jfa.2011.10.007.  Google Scholar

[5]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95.  Google Scholar

[6]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.  Google Scholar

[7]

R.-J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413. doi: 10.1142/S0219891611002421.  Google Scholar

[8]

R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, preprint, 2012. Google Scholar

[9]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint, 2012. Google Scholar

[10]

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

[11]

R.-J. Duan, T. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, to appear in Math. Models Methods Appl. Sci. (2012), arXiv:1112.1453v1. Google Scholar

[12]

R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint 2011, arXiv:1112.3261v1. Google Scholar

[13]

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

[14]

R.-J. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64 (2011), 1497-1546.  Google Scholar

[15]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.  Google Scholar

[16]

H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," Springer-Berlin, XII (1958), 205-294.  Google Scholar

[17]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.  Google Scholar

[18]

P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384. doi: 10.1016/j.aim.2011.05.005.  Google Scholar

[19]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.  Google Scholar

[20]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812. doi: 10.1090/S0894-0347-2011-00722-4.  Google Scholar

[21]

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

[22]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16 (2006), 1839-1859. doi: 10.1142/S021820250600173X.  Google Scholar

[23]

N. A. Krall and A. W. Trivelpiece, "Principles of Plasma Physics," McGraw-Hill, 1973. Google Scholar

[24]

P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), 191-204. doi: 10.1098/rsta.1994.0018.  Google Scholar

[25]

S.-Q. Liu and H.-J. Yu, Optimal time decay of the Landau equation with potential force, preprint, 2011. Google Scholar

[26]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar

[27]

T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011.  Google Scholar

[28]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348. doi: 10.1080/03605300600635004.  Google Scholar

[29]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567. doi: 10.1007/s00220-006-0109-y.  Google Scholar

[30]

R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinetic and Related Models, 5 (2012), 583-613. doi: 10.3934/krm.2012.5.583.  Google Scholar

[31]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.  Google Scholar

[32]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.  Google Scholar

[33]

R. M. Strain and Y. Guo, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Commun. Math. Phys., 10 (2012), 649-673. doi: 10.1007/s00220-012-1417-z.  Google Scholar

[34]

R. M. Strain and K. Zhu, The Vlasov-Poisson-Landau system in $\mathbbR^3$, preprint 2012, arXiv:1202.2471v1. Google Scholar

[35]

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

[36]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305.  Google Scholar

[37]

Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbbR^3$, preprint 2012, arXiv:1205.6277. Google Scholar

[38]

M.-Q. Zhan, Local existence of solutions to the Landau-Maxwell system, Math. Methods Appl. Sci., 17 (1994), 613-641. doi: 10.1002/mma.1670170804.  Google Scholar

show all references

References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.  Google Scholar

[2]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123. doi: 10.1007/s00205-010-0290-1.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 263 (2012), 915-1010. doi: 10.1016/j.jfa.2011.10.007.  Google Scholar

[5]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95.  Google Scholar

[6]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.  Google Scholar

[7]

R.-J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413. doi: 10.1142/S0219891611002421.  Google Scholar

[8]

R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, preprint, 2012. Google Scholar

[9]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint, 2012. Google Scholar

[10]

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

[11]

R.-J. Duan, T. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, to appear in Math. Models Methods Appl. Sci. (2012), arXiv:1112.1453v1. Google Scholar

[12]

R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint 2011, arXiv:1112.3261v1. Google Scholar

[13]

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

[14]

R.-J. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64 (2011), 1497-1546.  Google Scholar

[15]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.  Google Scholar

[16]

H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," Springer-Berlin, XII (1958), 205-294.  Google Scholar

[17]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.  Google Scholar

[18]

P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384. doi: 10.1016/j.aim.2011.05.005.  Google Scholar

[19]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.  Google Scholar

[20]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812. doi: 10.1090/S0894-0347-2011-00722-4.  Google Scholar

[21]

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

[22]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16 (2006), 1839-1859. doi: 10.1142/S021820250600173X.  Google Scholar

[23]

N. A. Krall and A. W. Trivelpiece, "Principles of Plasma Physics," McGraw-Hill, 1973. Google Scholar

[24]

P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), 191-204. doi: 10.1098/rsta.1994.0018.  Google Scholar

[25]

S.-Q. Liu and H.-J. Yu, Optimal time decay of the Landau equation with potential force, preprint, 2011. Google Scholar

[26]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar

[27]

T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011.  Google Scholar

[28]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348. doi: 10.1080/03605300600635004.  Google Scholar

[29]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567. doi: 10.1007/s00220-006-0109-y.  Google Scholar

[30]

R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinetic and Related Models, 5 (2012), 583-613. doi: 10.3934/krm.2012.5.583.  Google Scholar

[31]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.  Google Scholar

[32]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.  Google Scholar

[33]

R. M. Strain and Y. Guo, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Commun. Math. Phys., 10 (2012), 649-673. doi: 10.1007/s00220-012-1417-z.  Google Scholar

[34]

R. M. Strain and K. Zhu, The Vlasov-Poisson-Landau system in $\mathbbR^3$, preprint 2012, arXiv:1202.2471v1. Google Scholar

[35]

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

[36]

C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305.  Google Scholar

[37]

Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbbR^3$, preprint 2012, arXiv:1205.6277. Google Scholar

[38]

M.-Q. Zhan, Local existence of solutions to the Landau-Maxwell system, Math. Methods Appl. Sci., 17 (1994), 613-641. doi: 10.1002/mma.1670170804.  Google Scholar

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