# zbMATH — the first resource for mathematics

The Vlasov-Poisson-Landau system in a periodic box. (English) Zbl 1251.35167
Summary: 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 L. Landau in 1936 [Phys. Z. Sowjetunion 10, 154–164 (1936; Zbl 0015.38202)]. We construct global unique solutions to such a system for initial data which have small weighted $$H^{2}$$ 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.

##### MSC:
 35Q83 Vlasov equations 35Q84 Fokker-Planck equations 35C05 Solutions to PDEs in closed form
Full Text:
##### References:
 [1] A. A. Arsen$$^{\prime}$$ev and O. E. Buryak, On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation, Mat. Sb. 181 (1990), no. 4, 435 – 446 (Russian); English transl., Math. USSR-Sb. 69 (1991), no. 2, 465 – 478. · Zbl 0713.35075 [2] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 1, 61 – 95 (English, with English and French summaries). · Zbl 1044.83007 [3] Yemin Chen, Laurent Desvillettes, and Lingbing He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 21 – 55. · Zbl 1169.76064 [4] P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Rational Mech. Anal. 138 (1997), no. 2, 137 – 167. · Zbl 0888.35084 [5] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2005), no. 2, 245 – 316. · Zbl 1162.82316 [6] Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391 – 434. · Zbl 1042.76053 [7] Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002), no. 9, 1104 – 1135. · Zbl 1027.82035 [8] Yan Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math. 59 (2006), no. 5, 626 – 687. , https://doi.org/10.1002/cpa.20121 Yan Guo, Erratum: ”Boltzmann diffusive limit beyond the Navier-Stokes approximation” [Comm. Pure Appl. Math. 59 (2006), no. 5, 626 – 687; MR2172804], Comm. Pure Appl. Math. 60 (2007), no. 2, 291 – 293. · Zbl 1089.76052 [9] Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal. 169 (2003), no. 4, 305 – 353. · Zbl 1044.76056 [10] Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math. 153 (2003), no. 3, 593 – 630. · Zbl 1029.82034 [11] Mahir Hadžić and Yan Guo, Stability in the Stefan problem with surface tension (I), Comm. Partial Differential Equations 35 (2010), no. 2, 201 – 244. · Zbl 1195.35303 [12] Robert T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. · Zbl 0858.76001 [13] Guo, Y.; Tice, I.: Decay of viscous surface waves without surface tension. arXiv:1011.5179. · Zbl 1292.35206 [14] Guo, Y.; Strain, R.M.: Momentum Regularity and Stability of the Relativistic Vlasov-Maxwell-Boltzmann System. arXiv:1012.1158 · Zbl 1245.35130 [15] Gressman, T. P.; Strain, R. S.: Global classical solutions of the Boltzmann equation without angular cut-off. arXiv:1011.5441v1. · Zbl 1248.35140 [16] Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Natl. Acad. Sci. USA 107 (2010), no. 13, 5744 – 5749. · Zbl 1205.82120 [17] Hilton, F.: Collisional transport in plasma. Handbook of Plasma Physics. (1) Amsterdam: North-Holland, 1983. [18] Hadzic, M.: Orthogonality conditions and asymptotic stability in the Stefan problem with surface tension. arXiv:1101.5177 · Zbl 1132.49021 [19] Ling Hsiao and Hongjun Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math. 65 (2007), no. 2, 281 – 315. · Zbl 1143.35085 [20] P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 (1994), no. 1679, 191 – 204. · Zbl 0809.35137 [21] Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417 – 429. · Zbl 1096.82010 [22] Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287 – 339. · Zbl 1130.76069 [23] Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys. 251 (2004), no. 2, 263 – 320. · Zbl 1113.82070 [24] Cédric Villani, On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differential Equations 1 (1996), no. 5, 793 – 816. · Zbl 0856.35020 [25] Mei-Qin Zhan, Local existence of solutions to the Landau-Maxwell system, Math. Methods Appl. Sci. 17 (1994), no. 8, 613 – 641. · Zbl 0803.35114 [26] Mei-Qin Zhan, Local existence of classical solutions to the Landau equations, Transport Theory Statist. Phys. 23 (1994), no. 4, 479 – 499. · Zbl 0810.35095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.