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}, volume = {31}, year = {2014}, pages = {751-778}, doi = {10.1016/j.anihpc.2013.07.004}, mrnumber = {3249812}, zbl = {1305.82057}, language = {en}, url = {http://dml.mathdoc.fr/item/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, Tome 31 (2014) pp. 751-778. doi : 10.1016/j.anihpc.2013.07.004. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_4_751_0/
[1] On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 no. 1 (2004), 61 -95 | Numdam | MR 2037247 | Zbl 1044.83007
, ,[2] Dispersion relations for the linearized Fokker–Planck equation, Arch. Ration. Mech. Anal. 138 no. 2 (1997), 137 -167 | MR 1463805 | Zbl 0888.35084
, ,[3] On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 no. 2 (2005), 245 -316 | MR 2116276 | Zbl 1162.82316
, ,[4] Global smooth flows for the compressible Euler–Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ. 8 no. 2 (2011), 375 -413 | MR 2812147 | Zbl 1292.76080
,[5] The Vlasov–Poisson–Boltzmann system in the whole space: The hard potential case, J. Differential Equations 252 (2012), 6356 -6386 | MR 2911838 | Zbl 1247.35174
, , ,[6] The Vlasov–Poisson–Boltzmann system for soft potentials, Math. Models Methods Appl. Sci. 23 no. 6 (2013), 979 -1028 | MR 3037299 | Zbl 06183379
, , ,[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 2754344 | Zbl 1232.35169
, ,[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 2832167 | Zbl 1244.35010
, ,[10] The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), 391 -434 | MR 1946444 | Zbl 1042.76053
,[11] The Vlasov–Poisson–Landau system in a periodic box, J. Amer. Math. Soc. 25 (2012), 759 -812 | MR 2904573 | Zbl 1251.35167
,[12] The Vlasov–Maxwell–Boltzmann system near Maxwellians, Invent. Math. 153 no. 3 (2003), 593 -630 | MR 2000470 | Zbl 1029.82034
,[13] Collisional Transport in Magnetized Plasmas, Cambridge University Press (2002) | Zbl 1044.76001
, ,[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 2271601 | Zbl 1108.35014
, ,[15] On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math. 65 no. 2 (2007), 281 -315 | MR 2330559 | Zbl 1143.35085
, ,[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 1278244 | Zbl 0809.35137
,[18] Boltzmann equation: micro–macro decompositions and positivity of shock profiles, Commun. Math. Phys. 246 no. 1 (2004), 133 -179 | MR 2044894 | Zbl 1092.82034
, ,[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 2082240 | Zbl 1111.76047
, ,[20] Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations 31 (2006), 1321 -1348 | MR 2254617 | Zbl 1101.76053
,[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 2972454 | Zbl 06124540
,[22] Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), 287 -339 | MR 2366140 | Zbl 1130.76069
, ,[23] Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys. 251 no. 2 (2004), 263 -320 | MR 2100057 | Zbl 1113.82070
, ,[24] The Vlasov–Poisson–Landau system in , arXiv:1202.2471v1 (2012) | MR 3101794
, ,[25] On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad. 50 (1974), 179 -184 | MR 363332 | Zbl 0312.35061
,[26] A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam (2002), 71 -305 | MR 1942465 | Zbl 1170.82369
,[27] On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differential Equations 1 no. 5 (1996), 793 -816 | MR 1392006 | Zbl 0856.35020
,[28] Global classical solution of the Vlasov–Maxwell–Landau system near Maxwellians, J. Math. Phys. 45 no. 11 (2004), 4360 -4376 | MR 2098143 | Zbl 1064.82035
,[29] Local existence of solutions to the Landau–Maxwell system, Math. Methods Appl. Sci. 17 no. 8 (1994), 613 -641 | MR 1280648 | Zbl 0803.35114
,