The motion of a fully ionized plasma of electrons and ions is generally governed by the Vlasov–Maxwell–Landau system. We prove the global existence of solutions near Maxwellians to the Cauchy problem of the system for the long-range collision kernel of soft potentials, particularly including the classical Coulomb collision, provided that both the Sobolev norm and -norm of initial perturbation with enough smoothness and enough velocity weight is sufficiently small. As a byproduct, the convergence rates of solutions are also obtained. The proof is based on the energy method through designing a new temporal energy norm to capture different features of this complex system such as dispersion of the macro component in , singularity of the long-range collisions and regularity-loss of the electromagnetic field.
@article{AIHPC_2014__31_4_751_0, author = {Duan, Renjun}, title = {Global smooth dynamics of a fully ionized plasma with long-range collisions}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {751--778}, publisher = {Elsevier}, volume = {31}, number = {4}, year = {2014}, doi = {10.1016/j.anihpc.2013.07.004}, mrnumber = {3249812}, zbl = {1305.82057}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.004/} }
TY - JOUR AU - Duan, Renjun TI - Global smooth dynamics of a fully ionized plasma with long-range collisions JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 751 EP - 778 VL - 31 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.004/ DO - 10.1016/j.anihpc.2013.07.004 LA - en ID - AIHPC_2014__31_4_751_0 ER -
%0 Journal Article %A Duan, Renjun %T Global smooth dynamics of a fully ionized plasma with long-range collisions %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 751-778 %V 31 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.004/ %R 10.1016/j.anihpc.2013.07.004 %G en %F AIHPC_2014__31_4_751_0
Duan, Renjun. Global smooth dynamics of a fully ionized plasma with long-range collisions. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, pp. 751-778. doi : 10.1016/j.anihpc.2013.07.004. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.004/
[1] On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 no. 1 (2004), 61 -95 | EuDML | Numdam | MR | Zbl
, ,[2] Dispersion relations for the linearized Fokker–Planck equation, Arch. Ration. Mech. Anal. 138 no. 2 (1997), 137 -167 | MR | Zbl
, ,[3] On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 no. 2 (2005), 245 -316 | MR | Zbl
, ,[4] Global smooth flows for the compressible Euler–Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ. 8 no. 2 (2011), 375 -413 | MR | Zbl
,[5] The Vlasov–Poisson–Boltzmann system in the whole space: The hard potential case, J. Differential Equations 252 (2012), 6356 -6386 | MR | Zbl
, , ,[6] The Vlasov–Poisson–Boltzmann system for soft potentials, Math. Models Methods Appl. Sci. 23 no. 6 (2013), 979 -1028 | MR | Zbl
, , ,[7] Global solutions to the Vlasov–Poisson–Landau system, arXiv:1112.3261v1 (2011)
, , ,[8] Optimal time decay of the Vlasov–Poisson–Boltzmann system in , Arch. Ration. Mech. Anal. 199 no. 1 (2011), 291 -328 | MR | Zbl
, ,[9] Optimal large-time behavior of the Vlasov–Maxwell–Boltzmann system in the whole space, Comm. Pure Appl. Math. 64 no. 11 (2011), 1497 -1546 | MR | Zbl
, ,[10] The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), 391 -434 | MR | Zbl
,[11] The Vlasov–Poisson–Landau system in a periodic box, J. Amer. Math. Soc. 25 (2012), 759 -812 | MR | Zbl
,[12] The Vlasov–Maxwell–Boltzmann system near Maxwellians, Invent. Math. 153 no. 3 (2003), 593 -630 | MR | Zbl
,[13] Collisional Transport in Magnetized Plasmas, Cambridge University Press (2002) | Zbl
, ,[14] Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci. 16 no. 11 (2006), 1839 -1859 | MR | Zbl
, ,[15] On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math. 65 no. 2 (2007), 281 -315 | MR | Zbl
, ,[16] Principles of Plasma Physics, McGraw–Hill (1973)
, ,[17] On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 no. 1679 (1994), 191 -204 | MR | Zbl
,[18] Boltzmann equation: micro–macro decompositions and positivity of shock profiles, Commun. Math. Phys. 246 no. 1 (2004), 133 -179 | MR | Zbl
, ,[19] The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), 1543 -1608 | MR | Zbl
, ,[20] Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations 31 (2006), 1321 -1348 | MR | Zbl
,[21] Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinet. Relat. Models 5 no. 3 (2012), 583 -613 | MR | Zbl
,[22] Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), 287 -339 | MR | Zbl
, ,[23] Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys. 251 no. 2 (2004), 263 -320 | MR | Zbl
, ,[24] The Vlasov–Poisson–Landau system in , arXiv:1202.2471v1 (2012) | MR
, ,[25] On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad. 50 (1974), 179 -184 | MR | Zbl
,[26] A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam (2002), 71 -305 | MR | Zbl
,[27] On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differential Equations 1 no. 5 (1996), 793 -816 | MR | Zbl
,[28] Global classical solution of the Vlasov–Maxwell–Landau system near Maxwellians, J. Math. Phys. 45 no. 11 (2004), 4360 -4376 | MR | Zbl
,[29] Local existence of solutions to the Landau–Maxwell system, Math. Methods Appl. Sci. 17 no. 8 (1994), 613 -641 | MR | Zbl
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