TIME-PERIODIC SOLUTIONS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM

doi:10.1016/S0252-9602(15)30026-6

Abstract

Abstract:

In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin's method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.

Key words: Vlasov-Poisson-Fokker-Planck system, time-periodic solution, energy method, exponential time-decay

CLC Number:

35Q84

Renjun DUAN, Shuangqian LIU. TIME-PERIODIC SOLUTIONS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM[J].Acta mathematica scientia,Series B, 2015, 35(4): 876-886.

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References

[1] Bouchut F. Hypoelliptic regularity in kinetic equations. J Math Pures Appl, 2002, 81(9): 1135-1159
[2] Bouchut F. Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions. J Funct Anal, 1993, 111: 239-258
[3] Bouchut F. Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system. J Differential Equa- tions, 1995, 122: 225-238
[4] Bouchut F, Dolbeault J. On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov- Poisson-Fokker-Planck system with Coulombic and Newtonian potentials. Differential Integral Equations, 1995, 8: 487-514
[5] Carpio A. Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation. Math Methods Appl Sci, 1998, 21: 985-1014
[6] Carrillo J, Duan R, Moussa A. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck- Euler system. Kinet Relat Models, 2011, 4: 227-258
[7] Carrillo J A, Soler J. On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in Lp spaces. Math Methods Appl Sci, 1995, 18: 825-839
[8] Carrillo J A, Soler J, Vazquez J L. Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system. J Funct Anal, 1996, 141: 99-132
[9] Desvillettes L, Villani C. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation. Comm Pure Appl Math, 2001, 54: 1-42
[10] Desvillettes L, Villani C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent Math, 2005, 159: 245-316
[11] Duan R -J. Hypocoercivity of linear degenerately dissipative kinetic equations. Nonlinearity, 2011, 24(8): 2165-2189
[12] Duan R -J, Fornasier M, Toscani G. A kinetic flocking model with diffusion. Comm Math Phys, 2010, 300(1): 95-145
[13] Duan R -J, Liu S Q. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinet Relat Models, 2013, 6(4): 687-700
[14] Duan R -J, Strain R. Optimal time decay of the Vlasov-Poisson-Boltzmann system in R³. Arch Ration Mech Anal, 2011, 199: 291-328
[15] Duan R -J, Ukai S, Yang T, Zhao H J. Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications. Comm Math Phys, 2008, 277(1): 189-236
[16] Duan R -J, Yang T. Stability of the one-species Vlasov-Poisson-Boltzmann system. SIAM J Math Anal, 2009/10, 41(6): 2353-2387
[17] Esposito R, Guo Y, Marra R. Stability of the front under a Vlasov-Fokker-Planck dynamics. Arch Ration Mech Anal, 2010, 195: 75-116
[18] Guo Y. The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm Pure Appl Math, 2002, 55: 1104- 1135
[19] Guo Y. The Vlasov-Poisson-Landau system in a periodic box. J Amer Math Soc, 2012, 25: 759-812
[20] Hérau F, Nier F. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch Ration Mech Anal, 2004, 171: 151-218
[21] Hwang H J, Jang J. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete Contin Dyn Syst Ser B, 2013, 18: 681-691
[22] Maremonti P. Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space. Nonlinearity, 1991, 4: 503-529
[23] Ono K, Strauss W A. Regular solutions of the Vlasov-Poisson-Fokker-Planck system. Discrete Contin Dynam Systems, 2000, 6: 751-772
[24] Rein G, Weckler J. Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions. J Differential Equations, 1992, 99: 59-77
[25] Serrin J. A note on the existence of periodic solutions of the Navier-Stokes equations. Arch. Rational Mech Anal, 1959, 3: 120-122
[26] Ukai S. Time-periodic solutions of the Boltzmann equation. Discrete Cont Dyn Syst, 2006, 14: 579-596
[27] Ukai S, Yang T. The Boltzmann equation in the space L² ∩ L_β^∞ : Global and time-periodic solutions. Anal Appl, 2006, 4: 263-310
[28] Villani C. Hypocoercivity. Mem Amer Math Soc, 2009, 202(950)
[29] Victory H D. On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems. J Math Anal Appl, 1991, 160: 525-555
[30] Victory Jr H D, O'Dwyer B P. On classical solutions of Vlasov-Poisson Fokker-Planck systems. Indiana Univ Math J, 1990, 39: 105-156