×

zbMATH — the first resource for mathematics

Global smooth dynamics of a fully ionized plasma with long-range collisions. (English) Zbl 1305.82057
The author describes the dynamics of a fully ionized, two-component plasma under the influence of the self-consistent Lorentz force and binary collisions (taken into account by the Landau collision operator) by a system of kinetic transport equations and the Maxwell equations, which he calls “Vlasov-Maxwell-Landau system”. The main aim of the paper is the prove the global existence of solutions of the Cauchy problem of the Vlasov-Maxwell-Landau system of equations in the form of a disturbed global Maxwellian velocity distribution under some conditions on the initial data. In doing so, the long-range collisional kernel of soft potentials, particularly including the classical Coulomb collisions, is taken into account – assuming that both the Sobolev norm and the \(L^2_\xi(L^1_x)\)-norm of the initial perturbation “with enough smoothness and enough velocity weight” are sufficiently small. As a byproduct, also the convergence rates of the solutions are obtained. The proof is based on the energy method through designing a new temporal energy norm to capture different features of the complex system such as dispersion of the macro component in the velocity space, singularity of the long-range collisions and regularity-loss of the electromagnetic field.

MSC:
82D10 Statistical mechanics of plasmas
35Q84 Fokker-Planck equations
35Q83 Vlasov equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alexandre, R.; Villani, C., On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21, 1, 61-95, (2004) · Zbl 1044.83007
[2] Degond, P.; Lemou, M., Dispersion relations for the linearized Fokker-Planck equation, Arch. Ration. Mech. Anal., 138, 2, 137-167, (1997) · Zbl 0888.35084
[3] Desvillettes, L.; Villani, C., On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math., 159, 2, 245-316, (2005) · Zbl 1162.82316
[4] Duan, R.-J., Global smooth flows for the compressible Euler-Maxwell system: relaxation case, J. Hyperbolic Differ. Equ., 8, 2, 375-413, (2011) · Zbl 1292.76080
[5] Duan, R.-J.; Yang, T.; Zhao, H.-J., The Vlasov-Poisson-Boltzmann system in the whole space: the hard potential case, J. Differential Equations, 252, 6356-6386, (2012) · Zbl 1247.35174
[6] Duan, R.-J.; Yang, T.; Zhao, H.-J., The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23, 6, 979-1028, (2013) · Zbl 1437.76056
[7] Duan, R.-J.; Yang, T.; Zhao, H.-J., Global solutions to the Vlasov-Poisson-Landau system, (2011), unpublished note
[8] Duan, R.-J.; Strain, R. M., Optimal time decay of the Vlasov-Poisson-Boltzmann system in \(\mathbf{R}^3\), Arch. Ration. Mech. Anal., 199, 1, 291-328, (2011) · Zbl 1232.35169
[9] Duan, R.-J.; Strain, R. M., Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64, 11, 1497-1546, (2011) · Zbl 1244.35010
[10] Guo, Y., The Landau equation in a periodic box, Comm. Math. Phys., 231, 391-434, (2002) · Zbl 1042.76053
[11] Guo, Y., The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25, 759-812, (2012) · Zbl 1251.35167
[12] Guo, Y., The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. Math., 153, 3, 593-630, (2003) · Zbl 1029.82034
[13] Helander, P.; Sigmar, D. J., Collisional transport in magnetized plasmas, (2002), Cambridge University Press · Zbl 1044.76001
[14] Hosono, T.; Kawashima, S., Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16, 11, 1839-1859, (2006) · Zbl 1108.35014
[15] Hsiao, L.; Yu, H., On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65, 2, 281-315, (2007) · Zbl 1143.35085
[16] Krall, N. A.; Trivelpiece, A. W., Principles of plasma physics, (1973), McGraw-Hill
[17] Lions, P.-L., On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A, 346, 1679, 191-204, (1994) · Zbl 0809.35137
[18] Liu, T.-P.; Yu, S.-H., Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246, 1, 133-179, (2004) · Zbl 1092.82034
[19] Liu, T.-P.; Yu, S.-H., The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57, 1543-1608, (2004) · Zbl 1111.76047
[20] Mouhot, C., Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31, 1321-1348, (2006) · Zbl 1101.76053
[21] Strain, R. M., Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models, 5, 3, 583-613, (2012) · Zbl 1383.76414
[22] Strain, R. M.; Guo, Y., Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187, 287-339, (2008) · Zbl 1130.76069
[23] Strain, R. M.; Guo, Y., Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys., 251, 2, 263-320, (2004) · Zbl 1113.82070
[24] Strain, R. M.; Zhu, K., The Vlasov-Poisson-Landau system in \(\mathbb{R}^3\), (2012), preprint
[25] Ukai, S., On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50, 179-184, (1974) · Zbl 0312.35061
[26] Villani, C., A review of mathematical topics in collisional kinetic theory, (Handbook of Mathematical Fluid Dynamics, vol. I, (2002), North-Holland Amsterdam), 71-305 · Zbl 1170.82369
[27] Villani, C., On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differential Equations, 1, 5, 793-816, (1996) · Zbl 0856.35020
[28] Yu, H.-J., Global classical solution of the Vlasov-Maxwell-Landau system near maxwellians, J. Math. Phys., 45, 11, 4360-4376, (2004) · Zbl 1064.82035
[29] Zhan, M.-Q., Local existence of solutions to the Landau-Maxwell system, Math. Methods Appl. Sci., 17, 8, 613-641, (1994) · Zbl 0803.35114
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.