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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95.
[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.
[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.
[8]	R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, preprint, 2012.
[9]	R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint, 2012.
[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.
[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.
[12]	R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint 2011, arXiv:1112.3261v1.
[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.
[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.
[15]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[16]	H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," Springer-Berlin, XII (1958), 205-294.
[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.
[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.
[19]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.
[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.
[21]	Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.
[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.
[23]	N. A. Krall and A. W. Trivelpiece, "Principles of Plasma Physics," McGraw-Hill, 1973.
[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.
[25]	S.-Q. Liu and H.-J. Yu, Optimal time decay of the Landau equation with potential force, preprint, 2011.
[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.
[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.
[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.
[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.
[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.
[31]	R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.
[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.
[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.
[34]	R. M. Strain and K. Zhu, The Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1202.2471v1.
[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.
[36]	C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305.
[37]	Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1205.6277.
[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.

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.
[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.
[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.
[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.
[5]	R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95.
[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.
[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.
[8]	R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, preprint, 2012.
[9]	R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint, 2012.
[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.
[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.
[12]	R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint 2011, arXiv:1112.3261v1.
[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.
[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.
[15]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[16]	H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," Springer-Berlin, XII (1958), 205-294.
[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.
[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.
[19]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.
[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.
[21]	Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.
[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.
[23]	N. A. Krall and A. W. Trivelpiece, "Principles of Plasma Physics," McGraw-Hill, 1973.
[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.
[25]	S.-Q. Liu and H.-J. Yu, Optimal time decay of the Landau equation with potential force, preprint, 2011.
[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.
[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.
[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.
[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.
[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.
[31]	R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545.
[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.
[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.
[34]	R. M. Strain and K. Zhu, The Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1202.2471v1.
[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.
[36]	C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305.
[37]	Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1205.6277.
[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.

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