The Vlasov-Poisson-Landau system in a periodic box

Author:
Yan Guo

Journal:
J. Amer. Math. Soc. **25** (2012), 759-812

MSC (2010):
Primary 35-XX

DOI:
https://doi.org/10.1090/S0894-0347-2011-00722-4

Published electronically:
October 25, 2011

MathSciNet review:
2904573

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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Vlasov-Poisson-Landau system describes the dynamics of a collisional plasma interacting with its own electrostatic field as well as its grazing collisions. Such grazing collisions are modeled by the famous Landau (Fokker-Planck) collision kernel, proposed by Landau in 1936. We construct global unique solutions to such a system for initial data which have small weighted norms, but can have large high derivatives with high velocity moments. Our construction is based on the accumulative study of the Landau kernel in the past decade, with four extra ingredients to overcome the specific mathematical difficulties present in the Vlasov-Poisson-Landau system: a new exponential weight of electric potential to cancel the growth of the velocity, a new velocity weight to capture the weak velocity diffusion in the Landau kernel, a decay of the electric field to close the energy estimate, and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude.

**[AB]**Arsen'ev, A. A.; Buryak, O.E.: On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation. Math. USSR. Sbornik 69 (2), 465-478 (1991). MR**1055522 (91f:35266)****[AV]**Alexandre, R.; Villani, C.; On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 1, 61-95. MR**2037247 (2005f:82126)****[CDH]**Chen, Y.; Desvillettes, L.; He, L.: Smoothing effects for classical solutions of the full Landau equation. Arch. Ration. Mech. Anal. 193 (2009), no. 1, 21-55, MR**2506070 (2010m:35062)****[DL]**Degond, P.; Lemou, M.: Dispersion relations for the linearized Fokker-Planck equation. Arch. Rational Mech. Anal. 138 (2), 137-167 (1997). MR**1463805 (99f:82051)****[DV]**Desvillettes, L.; Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159 (2005), no. 2, 245-316. MR**2116276 (2005j:82070)****[G1]**Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys., 231 (2002) 3, 391-434. MR**1946444 (2004c:82121)****[G2]**Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math., Vol. LV, (2002), 1104-1135. MR**1908664 (2003b:82050)****[G3]**Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. 59 (2006), no. 5, 626-687. (Erratum) 60 (2007), no. 2, 291-293. MR**2172804 (2007b:35047)**; MR**2275331 (2007g:35014)****[G4]**Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Ration. Mech. Anal. 169 (2003), no. 4, 305-353. MR**2013332 (2004i:82054)****[G5]**Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153 (2003), no. 3, 593-630. MR**2000470 (2004m:82123)****[GH]**Guo, Y.; Hadzic, M.: Stability of Stefan problem with surface tension (I). Comm. PDE, 1532-4133, 35, 2, (2010), 201-244. MR**2748623 (2011k:35255)****[GL]**Glassey, R.: The Cauchy Problems in Kinetic Theory, SIAM, 1996. MR**1379589 (97i:82070)****[GT]**Guo, Y.; Tice, I.: Decay of viscous surface waves without surface tension. arXiv:1011.5179.**[GS]**Guo, Y.; Strain, R.M.: Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System. arXiv:1012.1158**[GrS1]**Gressman, T. P.; Strain, R. S.: Global classical solutions of the Boltzmann equation without angular cut-off. arXiv:1011.5441v1.**[GrS2]**Gressman, T. P.; Strain, R. S.: Global Classical solutions of the Boltzmann equation with Long-Range interactions. Proc. Nat. Acad. Sci. U. S. A. March 30, 2010; 107 (13), 5744-5749. MR**2629879 (2011c:82064)****[H]**Hilton, F.: Collisional transport in plasma. Handbook of Plasma Physics. (1) Amsterdam: North-Holland, 1983.**[Ha]**Hadzic, M.: Orthogonality conditions and asymptotic stability in the Stefan problem with surface tension. arXiv:1101.5177**[HY]**Hsiao, L.; Yu, H.: On the Cauchy problem of the Boltzmann and Landau equations with soft potentials. Quart. Appl. Math. 65 (2007), no. 2, 281-315. MR**2330559 (2008i:35235)****[L]**Lions, P-L.: On Boltzmann and Landau equations. Phil. Trans. R. Soc. Lond. A 346, 191-204 (1994). MR**1278244 (95d:82050)****[SG1]**Strain, R. M.; Guo, Y.: Almost exponential decay near Maxwellian. Comm. Partial Differential Equations 31 (2006), no. 1-3, 417-429. MR**2209761 (2006m:82042)****[SG2]**Strain, R. M.; Guo, Y.: Exponential decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287-339. MR**2366140 (2008m:82008)****[SG3]**Strain, R. M.; Guo, Y.: Stability of the relativistic Maxwellian in a collisional plasma. Comm. Math. Phys. 251 (2004), no. 2, 263-320. MR**2100057 (2005m:82155)****[V]**Villani, C.: On the Landau equation: Weak stability, global existence. Adv. Diff. Eq. 1 (1996) (5), 793-816. MR**1392006 (97e:82048)****[Z1]**Zhan, M.-Q.: Local existence of solutions to the Landau-Maxwell system. Math. Methods Appl. Sci. 17 (1994), no. 8, 613-641. MR**1280648 (95h:35228)****[Z2]**Zhan, M.-Q.: Local existence of classical solutions to the Landau equations. Transport Theory Statist. Phys. 23 (1994), no. 4, 479-499. MR**1264848 (95c:35203)**

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Additional Information

**Yan Guo**

Affiliation:
Division of Applied Mathematics, Brown University, Box F, Providence, Rhode Island 02912

DOI:
https://doi.org/10.1090/S0894-0347-2011-00722-4

Received by editor(s):
March 19, 2011

Received by editor(s) in revised form:
June 27, 2011, and September 3, 2011

Published electronically:
October 25, 2011

Additional Notes:
This research is supported in part by NSF grant #0905255 and FRG grants as well as a Chinese NSF grant #10828103.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.