# American Institute of Mathematical Sciences

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

 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.
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 $\mathbbR^3$, 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 $\mathbbR^3$, 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 $\mathbbR^3$, 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 $\mathbbR^3$, 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.
 [1] Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic and Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005 [2] Yuanjie Lei, Huijiang Zhao. The Vlasov-Maxwell-Boltzmann system near Maxwellians with strong background magnetic field. Kinetic and Related Models, 2020, 13 (3) : 599-621. doi: 10.3934/krm.2020020 [3] Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299 [4] Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic and Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919 [5] Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic and Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036 [6] Nadia Lekrine, Chao-Jiang Xu. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation. Kinetic and Related Models, 2009, 2 (4) : 647-666. doi: 10.3934/krm.2009.2.647 [7] Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic and Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015 [8] Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020 [9] Sergiu Klainerman, Gigliola Staffilani. A new approach to study the Vlasov-Maxwell system. Communications on Pure and Applied Analysis, 2002, 1 (1) : 103-125. doi: 10.3934/cpaa.2002.1.103 [10] Jonathan Ben-Artzi, Stephen Pankavich, Junyong Zhang. A toy model for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2022, 15 (3) : 341-354. doi: 10.3934/krm.2021053 [11] Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic and Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003 [12] Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153 [13] Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040 [14] Yemin Chen. Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 889-910. doi: 10.3934/dcds.2008.20.889 [15] Dayton Preissl, Christophe Cheverry, Slim Ibrahim. Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system. Kinetic and Related Models, 2021, 14 (6) : 1035-1079. doi: 10.3934/krm.2021042 [16] Radjesvarane Alexandre, Mouhamad Elsafadi. Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations II. Non cutoff case and non Maxwellian molecules. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 1-11. doi: 10.3934/dcds.2009.24.1 [17] Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457 [18] Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 [19] Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 [20] Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253

2020 Impact Factor: 1.432