The pointwise space-time behaviors of the Green's function and the global solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system in three dimensional space are studied in this paper. It is shown that the Green's function consists of the diffusion waves decaying exponentially in time but algebraically in space, and the singular kinetic waves which become smooth for all $(t,x,v)$ when $t>0.$ Furthermore, we establish the pointwise space-time behaviors of the global solution to the nonlinear VPFP system when the initial data is not necessarily smooth in terms of the Green's function.
Mingying zhong
. GREEN'S FUNCTION AND THE POINTWISE BEHAVIORS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM*[J]. Acta mathematica scientia, Series B, 2023
, 43(1)
: 205
-236
.
DOI: 10.1007/s10473-023-0113-8
[1] Bonilla L, Carrillo J A, Soler J.Asymptotic behavior of an initial-boundary value problem for the Vlasov- Poisson-Fokker-Planck system. SIAM J Appl Math, 1997, 57(5): 1343-1372
[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(1): 239-258
[3] Bouchut F.Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system. J Differ Equ, 1995, 122(2): 225-238
[4] Bouchut F, Dolbeault J.On long asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov- Poisson-Fokker-Planck system with coulombic and newtonian potentials. Differ Integral Equ, 1995, 8: 487-514
[5] Carrillo J A.Global weak solutions for the initial-boundary value problems to the Vlasov-Poisson-Fokker- Planck system. Math Methods Appl Sci, 1998, 21: 907-938
[6] 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(10): 825-839
[7] 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
[8] El Ghani N, Masmoudi N.Diffusion limit of the Vlasov-Poisson-Fokker-Planck system. Commun Math Sci, 2010, 8(2): 463-479
[9] Goudon T.Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: analysis of the two- dimensional case. Math Models Methods Appl Sci, 2005, 15(5): 737-752
[10] Goudon T, Nieto J, Poupaud F, Soler J.Multidimensional high-field limit of the electro-static Vlasov- Poisson-Fokker-Planck system. J Differ Equ, 2005, 213(2): 418-442
[11] Hwang H J, Jang J.On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete Contin Dyn Syst Ser B, 2013, 18(3): 681-691
[12] Kolmogorov A.Zufallige Bewgungen (zur Theorie der Brownschen Bewegung). Ann of Math, 1934, 35: 116-117
[13] Li H L, Yang T, Sun J W, Zhong M.Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker- Planck) equations
(in Chinese). Sci Sin Math, 2016, 46: 981-1004
[14] Li H L, Yang T, Zhong M.Green’s function and pointwise space-time behaviors of the Vlasov-Poisson- Boltzmann system. Arch Rational Mech Anal, 2020, 235: 1011-1057
[15] Lin Y C, Wang H, Wu K C.Explicit structure of the Fokker-Planck equation with potential. Quart Appl Math, 2019, 77: 727-766
[16] Lin Y C, Wang H, Wu K C.Quantitative pointwise estimate of the solution of the linearized Boltzmann equation. J Stat Phys, 2018, 171: 927-964
[17] 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, 2004, 57: 1543-1608
[18] Liu T P, Yu S H.The Green’s function of Boltzmann equation, 3D waves. Bull Inst Math Acad Sin (NS), 2006, 1(1): 1-78
[19] Liu T P, Yu S H.Solving Boltzmann equation, Part I: Green’s function. Bull Inst Math Acad Sin (NS), 2011, 6: 151-243
[20] Luo L, Yu H J.Spectrum analysis of the linear Fokker-Planck equation. Anal Appl, 2017, 15(3): 313-331
[21] Markowich P A, Ringhofer C A, Schmeiser C. Semiconductor Equations.New York: Springer-Verlag, Wien, 1990
[22] Nieto J, Poupaud F, Soler J.High-field limit for the Vlasov-Poisson-Fokker-Planck system. Arch Ration Mech Anal, 2001, 158(1): 29-59
[23] Poupaud F, Soler J.Parabolic limit and stability of the Vlasov-Fokker-Planck system. Math Models Meth- ods Appl Sci, 2000, 10(7): 1027-1045
[24] Rein G, Weckler J.Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions. J Differ Equ, 1992, 99(1): 59-77
[25] Tanski I A.Fundamental solution of Fokker-Planck equation. arXiv:nlin/0407007, 2004
[26] Tanski I A. Fundamental solution of degenerated Fokker-Planck equation. arXiv:0804.0303, 2008
[27] Wu H, Lin T C, Liu C.Diffusion limit of kinetic equations for multiple species charged particles. Arch Rational Mech Anal, 2015, 215: 419-441
[28] Victory H D.On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems. J Math Anal Appl, 1991, 160(2): 525-555
[29] Victory H D, O’Dwyer B P. On classical solutions of Vlasov-Poisson-Fokker-Planck systems. Indiana Univ Math J, 1990, 39(1): 105-156
[30] Wang W, Wu Z.Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi- dimensions. J Diff Equs, 2010, 248: 1617-1636
[31] Wang W, Wu Z.Pointwise estimates of solution for non-isentoopic Navier-Stokes-Poisson equations in multi-dimensions. Acta Math Sci, 2012, 32B(5): 1681-1702
