Vlasov-Poisson-Boltzmann 方程组的 Navier-Stokes-Poisson 方程组逼近——献给陈化教授 70 寿辰

Abstract

Abstract:

When the Knudsen number approaches zero, the compressible Navier-Stokes-Poisson (NSP) system is not the limit of the dimensionless Vlasov-Poisson-Boltzmann (VPB) system; however, via the Chapman-Enskog expansion, it constitutes a second-order approximation to the VPB system. The purpose of this paper is to rigorously prove that if the difference between the local Maxwellian corresponding to the compressible NSP system and the initial value of the VPB system is a second-order small quantity in the Knudsen number, then the solutions of the two systems will maintain this order of magnitude difference for all times. The analysis in this paper is based on the energy estimates of the macroscopic equations, as well as the refined energy method within the macroscopic-microscopic decomposition framework.

Key words: Vlasov-Poisson-Boltzmann system, macroscopic-microscopic decomposition, compressible Navier-Stokes-Poisson system

CLC Number:

O175.2

Lina Dong, Shuangqian Liu, Xuan Ma, Yuanyuan Ma. Compressible Navier-Stokes-Poisson System Approximation to the Vlasov-Poisson-Boltzmann System[J].Acta mathematica scientia,Series A, 2026, 46(2): 683-708.

Trendmd

References 28

[1]	Bardos C, Golse F, Levermore D. Fluid dynamic limits of kinetic equations. I. Formal derivations. J Statist Phys, 1991, 63(1/2): 323-344
[2]	Bardos C, Golse F, Levermore D. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm Pure Appl Math, 1993, 46(5): 667-753
[3]	Cercignani C, Illner R, Pulvirenti M. The Mathematical Theory of Dilute Gases. Volume 106 of Applied Mathematical Sciences. New York: Springer-Verlag, 1994
[4]	Duan R J, Liu S Q. Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system. SIAM J Math Anal, 2015, 47(5): 3585-3647
[5]	Duan R J, Liu S Q. Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains. Trans Amer Math Soc, 2021, 374(11): 7867-7924
[6]	Duan R J, Yang D C, Yu H J. Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system. Math Models Methods Appl Sci, 2023, 33(10): 2157-2221
[7]	Golse F, Perthame B, Sulem C. On a boundary layer problem for the nonlinear Boltzmann equation. Arch Rational Mech Anal, 1988, 103(1): 81-96
[8]	Golse F, Saint-Raymond L. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent Math, 2004, 155(1): 81-161
[9]	Guo Y. Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm Pure Appl Math, 2006, 59(5): 626-687
[10]	Guo Y, Jang J. Global Hilbert expansion for the Vlasov-Poisson-Boltzmann system. Comm Math Phys, 2010, 299(2): 469-501
[11]	Guo Y, Jang J, Jiang N. Local Hilbert expansion for the Boltzmann equation. Kinet Relat Models, 2009, 2(1): 205-214
[12]	Guo Y, Liu S Q. Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation. Math Models Methods Appl Sci, 2017, 27(12): 2261-2296
[13]	Guo Y, Xiao Q H. Global Hilbert expansion for the relativistic Vlasov-Maxwell-Boltzmann system. Comm Math Phys, 2021, 384(1): 341-401
[14]	Huang F M, Wang Y, Wang Y, Yang T. The limit of the Boltzmann equation to the Euler equations for Riemann problems. SIAM J Math Anal, 2013, 45(3): 1741-1811
[15]	Huang F M, Xin Z P, Yang T. Contact discontinuity with general perturbations for gas motions. Adv Math, 2008, 219(4): 1246-1297
[16]	Jiang N, Masmoudi N. Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in bounded domain I. Comm Pure Appl Math, 2017, 70(1): 90-171
[17]	Lei Y J, Liu S Q, Xiao Q H, Zhao H J. Global Hilbert expansion for some non-relativistic kinetic equations. J Lond Math Soc, 2024, 110(5): Art e70016
[18]	Li F C, Wang Y C. Global Euler-Poisson limit to the Vlasov-Poisson-Boltzmann system with soft potential. SIAM J Math Anal, 2023, 55(4): 2877-2916
[19]	Li H L, Yang T, Zhong M Y. Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations. Indiana Univ Math J, 2016, 65(2): 665-725
[20]	Li H L, Yang T, Zhong M Y. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinet Relat Models, 2021, 14(2): 211-255
[21]	Liu S Q, Yang T, Zhao H J. Compressible Navier-Stokes approximation to the Boltzmann equation. J Differential Equations, 2014, 256(11): 3770-3816
[22]	Liu T P, Yang T, Yu S H. Energy method for Boltzmann equation. Phys D, 2004, 188(3/4): 178-192
[23]	Liu T P, Yang T, Yu S H, Zhao H J. Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch Ration Mech Anal, 2006, 181(2): 333-371
[24]	Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20(1): 67-104
[25]	Glassey R T. The Cauchy Problem in Kinetic Theory. Philadelphia, PA: SIAM, 1996
[26]	Xin Z P, Zeng H H. Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations. J Differential Equations, 2010, 249(4): 827-871
[27]	Yang T, Yu H J, Zhao H J. Cauchy problem for the Vlasov-Poisson-Boltzmann system. Arch Ration Mech Anal, 2006, 182(3): 415-470
[28]	Yang T, Zhao H J. A new energy method for the Boltzmann equation. J Math Phys, 2006, 47(5): Art 053301

Compressible Navier-Stokes-Poisson System Approximation to the Vlasov-Poisson-Boltzmann System

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 28

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Wenjie Zhang, Chunlai Mu. Global Boundedness and Large Time Behavior for a Chemotaxis-convection Model Describing Tumor Angiogenesis [J]. Acta mathematica scientia,Series A, 2026, 46(2): 415-427.
[2]	Yunlu Fan, Xin Liao. A Bahri-Lions Type Theorem for Dirichlet Forms and Its Applications to Nonlinear Degenerate Elliptic Equations [J]. Acta mathematica scientia,Series A, 2026, 46(2): 428-451.
[3]	Zhengzheng Chen, Dan Lei, Yuxin Yan, Huijiang Zhao. Stability of Stationary Solutions to the Non-Isentropic Compressible Navier-Stokes-Allen-Cahn Equations Under a Class of Large Initial Data [J]. Acta mathematica scientia,Series A, 2026, 46(2): 452-472.
[4]	Hui Liu. Some New Progress for the Problems About Closed Characteristics on Compact Star-Shaped Hypersurfaces [J]. Acta mathematica scientia,Series A, 2026, 46(2): 493-502.
[5]	Yawei Wei, Mengnan Zhang. Comparison Principle for Nonlinear Cone Degenerate Laplace Equations [J]. Acta mathematica scientia,Series A, 2026, 46(2): 503-517.
[6]	Yaping Guo, Jialin Li, Wenbin Lyu. Global Existence for a Class of Chemotaxis Systems with Signal-Dependent Motility and Generalized Logistic Source [J]. Acta mathematica scientia,Series A, 2026, 46(2): 535-551.
[7]	Weiwei Ao, Shanshan Lai. Sharp Estimates for Fully Bubbling Solutions of $G_2$ Toda System [J]. Acta mathematica scientia,Series A, 2026, 46(2): 552-583.
[8]	Yong Luo, Chunpeng Wang, Jingxue Yin. Flux-Limited Diffusion Equations and Their $BV$ Theory [J]. Acta mathematica scientia,Series A, 2026, 46(2): 604-615.
[9]	Weixi Li, Jiaxi Wang, Zhan Xu. Long-Time Well-Posedness of the Hyperbolic Prandtl Equations in Gevrey Spaces [J]. Acta mathematica scientia,Series A, 2026, 46(2): 616-627.
[10]	Lingjun Wang. Hydroelastic Small-Amplitude Solitary and Generalized Solitary Waves with Constant Vorticity [J]. Acta mathematica scientia,Series A, 2026, 46(2): 628-645.
[11]	Juncheng Wei, Yifu Zhou. Parabolic Gluing Method and Singularity Formation [J]. Acta mathematica scientia,Series A, 2026, 46(2): 646-668.
[12]	Hao Liu, Yun Wang, Chunjing Xie. Existence of Discretely Self-Similar Solutions to Four-Dimensional Steady Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2026, 46(2): 709-723.
[13]	Peng Luo, Keke Wang, Wenjie Wang. Existence of Positive Solutions to the Hénon Problem Without Compactness Conditions [J]. Acta mathematica scientia,Series A, 2026, 46(2): 724-736.
[14]	Jiaqing Bao, Nanbo Chen, Xiaochun Liu, Sainan Liang. On the Schrödinger-Bopp-Podolsky-Proca System with Singular Nonlinearity on Closed Manifolds [J]. Acta mathematica scientia,Series A, 2026, 46(2): 751-769.
[15]	Tianyi Wang, Zhengyang Yu. Rigidity of Steady Inhomogeneous Incompressible Euler Equations in Two-Dimensional Annular Domains [J]. Acta mathematica scientia,Series A, 2026, 46(2): 819-839.