STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

doi:10.1007/s10473-019-0112-y

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (1): 127-138.doi: 10.1007/s10473-019-0112-y

• Articles • Previous Articles Next Articles

STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

Jie LIAO¹, Qianrong WANG¹, Xiongfeng YANG²

1. Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China;
2. School of Mathematical Sciences;Key Laboratory of Scientific and Engineering Computing(MOE), Shanghai Jiao Tong University, Shanghai, 200240, China

Received:2017-09-15 Revised:2018-02-24 Online:2019-02-25 Published:2019-03-13
Contact: Jie LIAO E-mail:liaojie@ecust.edu.cn
Supported by:
The research was partially supported by Fundamental Research Funds for the Central Universities, NSFC (11871335) and by the SJTU's SMC Projection.

Abstract

Abstract: In this article, we establish the exponential time decay of smooth solutions around a global Maxwellian to the non-linear Vlasov—Poisson—Fokker—Planck equations in the whole space by uniform-in-time energy estimates. The non-linear coupling of macroscopic part and Fokker—Planck operator in the model brings new difficulties for the energy estimates, which is resolved by adding tailored weighted-in-v energy estimates suitable for the Fokker—Planck operator.

Key words: non-linear Vlasov-Poisson-Fokker-Planck equation, global Maxwellian, global a priori estimates, exponential convergence

Jie LIAO, Qianrong WANG, Xiongfeng YANG. STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS[J].Acta mathematica scientia,Series B, 2019, 39(1): 127-138.

Trendmd

References

[1] Chandrasekhar S. Stochastic problems in physics and astronomy. Reviews of Modern Physics, 1943, 15(1):1-89
[2] Hwang H J, Jang J. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete and Continuous Dynamical Systems Series B, 2013, 18(3):681-691
[3] Jin S, Zhu Y. Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales. SIAM J Math Anal, 2018, 50(2):1790-1816
[4] Perthame B. Higher moments for kinetic equations:The Vlasov-Poisson and Fokker-Planck cases. Mathematical Methods in the Applied Sciences, 1990, 13:441-452
[5] Luo L, Yu H. Spectrum analysis of the linear Fokker-Planck equation. Analysis and Applications, 2017, 15:313-331
[6] Liao J, Wang Q, Yang X. Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations. J Stat Phys, 2018, 173(1):222-241
[7] Galkin V S, Rusakov S V. Kinetic Fokker-Planck equation for free-molecular, thermally nonequilibrium Brownian particles in an inhomogeneous gas. Fluid Dynamics, 2007, 42(2):330-334
[8] Montgomery D. Brownian motion from Boltzmann's equation. The Physics of Fluids, 1971, 14:2088-2090
[9] Villani C. A review of mathematical topics in collisional kinetic theory//Handbook of Mathematical Fluid Dynamics. Amsterdam:North-Holland, Vol I. 2002:71-305
[10] Frank T D. Nonlinear Fokker-Planck Equations:Fundamentals and Applications. Springer, 2005
[11] Carrillo J A, Toscani G. Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations. Mathematical Methods in the Applied Sciences, 1998, 21(13):1269-1286
[12] Yang T, Yu H. Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system. SIAM J Math Anal, 2010, 42(1):459-488
[13] Carrillo J A, Duan R, Moussa A. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinetic and Related Models, 2011, 4(1):227-258
[14] Goudon T, He L, Moussa A, et al. The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium. SIAM J Math Anal, 2009, 42(5):2177-2202
[15] Li F, Mu Y, Wang D. Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck Equations:Global existence near the equilibrium and large time behavior. SIAM J Math Anal, 2017, 49(2):984-1026
[16] Mellet A, Vasseur A. Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Comm Math Phys, 2008, 281:573-596
[17] Villani C. Hypocoercivity. Mem Amer Math Soc, 2009, 202(950)
[18] Hörmander L. Hypoelliptic second order differential equations. Acta Math, 1967, 119:147-171
[19] Bouchut F. Hypoelliptic regularity in kinetic equations. J Math Pures Appl, 2002, 81(11):1135-1159
[20] Stein E. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1971

STABILITY OF GLOBAL MAXWELLIAN FOR NON-LINEAR VLASOV-POISSON-FOKKER-PLANCK EQUATIONS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 0

Metrics

Comments

Recommended 10