MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (5): 1549-1566.

MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION

黄飞敏, 王勇

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;Academy of Mathematics and System Sciences, CAS, Beijing 100190, China

收稿日期:2018-02-01 出版日期:2018-11-09 发布日期:2018-11-09
通讯作者: Yong WANG,E-mail:yongwang@amss.ac.cn E-mail:yongwang@amss.ac.cn
作者简介:Feimin HUANG,E-mail:fhuang@amt.ac.cn
基金资助:
Feimin Huang was partially supported by by National Center for Mathematics and Interdisciplinary Sciences, AMSS, CAS and NSFC (11371349 and 11688101). Yong Wang was partially supported by NSFC (11688101, 11771429).

MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION

Feimin HUANG, Yong WANG

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;Academy of Mathematics and System Sciences, CAS, Beijing 100190, China

Received:2018-02-01 Online:2018-11-09 Published:2018-11-09
Contact: Yong WANG,E-mail:yongwang@amss.ac.cn E-mail:yongwang@amss.ac.cn
Supported by:
Feimin Huang was partially supported by by National Center for Mathematics and Interdisciplinary Sciences, AMSS, CAS and NSFC (11371349 and 11688101). Yong Wang was partially supported by NSFC (11688101, 11771429).

摘要/Abstract

摘要： The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of (x,t) in the region R³×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.

关键词: Boltzmann equation, macroscopic regularity, compressible Navier-Stokes equations

Abstract: The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of (x,t) in the region R³×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.

Key words: Boltzmann equation, macroscopic regularity, compressible Navier-Stokes equations

黄飞敏, 王勇. MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION[J]. 数学物理学报(英文版), 2018, 38(5): 1549-1566.

Feimin HUANG, Yong WANG. MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION[J]. Acta Mathematica Scientia, 2018, 38(5): 1549-1566.

参考文献

[1] Alexandre R, Morimoto Y, Ukai S, Xu C-J, Yang T. Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff. Kyoto J Math, 2012, 52(3):433-463
[2] Alexandre R, Morimoto X, Ukai S, Xu C-J, Yang T. Regularizing effect and local existence for the non-cutoff Boltzmann equation. Arch Ration Mech Anal, 2010, 198(1):39-123
[3] Aoki K, Bardos C, Dogbe C, Golse F. A note on the propagation of boundary induced discontinuities in kinetic theory. Math Models Methods Appl Sci, 2001, 11(9):1581-1595
[4] Bardos C, Gamba I M, Franois G, Levermore C D. Global solutions of the Boltzmann equation over R^D near global Maxwellians with small mass. Comm Math Phys, 2016, 346(2):435-467
[5] Bellomo N, Palczewski A, Toscani G. Mathematical Topics in Nonlinear Kinetic Theory. Singapore:World Scientific Publishing Co, 1988
[6] Bellomo N, Toscani G. On the Cauchy problem for the nonlinear Boltzmann equation global existence uniqueness and asymptotic stability. J Math Phys, 1984, 26:334-338
[7] Bernis, L, Desvillettes L. Propagation of singularities for classical solutions of the Vlasov-PoissonBoltzmann equation. Discrete Contin Dyn Syst, 2009, 24:13-33
[8] Boudin L, Desvillettes L. On the singularities of the global small solution soft the full Boltzmann equation. Monatsch Math, 2000, 131:91-108
[9] Cannone M, Karch G. Infinite energy solutions to the homogeneous Boltzmann equation. Comm Pure Appl Math, 2010, 63:747-778
[10] Carlen E A, Gabetta E, Toscani G. Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Comm Math Phys, 1999, 199:521-546
[11] Cercignani C, Illner R, Pulvirenti M. The Mathematical Theory of Dilute Gases. New York:Springer-Verlag, 1994
[12] Desvillettes L, Wennberg B. Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm Partial Diff Eqns, 2004, 29:133-155
[13] DiPerna R J. Lions P-L. On the Cauchy problem for Boltzmann equation:Global existence and weak stability. Ann Math, 1989, 130:321-366
[14] Duan R J. On the Cauchy problem for the Boltzmann equation in the whole space:Global existence and uniform stability in L_ξ²(H_x^N). J Differential Equations, 2008, 244:3204-3234
[15] Duan R J, Huang F M, Wang Y, Yang T. Global well posedness for the Boltzmann equation with large amplitude initial data. Arch Rational Mech Anal, 2017, 225:375-424
[16] Duan R J, Li M R, Yang T. Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. Math Models Methods Appl Sci, 2008, 18:1093-1114
[17] Glassey R T. The Cauchy Problem in Kinetic Theory. Philadelphia:Society for Industrial and Applied Mathematics (SIAM), 1996
[18] Golse F, Lions P L, Perthame B, Sentis R. Regularity of the moments of the solution of a transport equation. J Funct Anal, 1988, 76(1):110125
[19] Guo Y. The Boltzmann equation in the whole space. Indiana Univ Math J, 2004, 53:1081-1094
[20] Guo Y. Bounded solutions for the Boltzmann equation. Quart Appl Math, 2010, 68(1):143-148
[21] Hamdache K. Existence in the large and asymptotic behaviour for the Boltzmann equation. Jpn J Appl Math, 1985, 2:1-15
[22] Hoff D, Liu T P. The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data. Indiana Univ Math J, 1989, 38(4):861-915
[23] Illner R, Shinbrot M. Global existence for a rare gas in an infinite vacuum. Comm Math Phys, 1984, 95:117-126
[24] Imai K, Nishida T. Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ RIMS Kyoto Univ, 1976, 12:229-239
[25] Kaniel S, Shinbrot M. The Boltzmann equation I:Uniqueness and global existence. Comm Math Phys, 1978, 58:65-84
[26] Kim C. Formation and propagation of discontinuity for Boltzmann equation in non-convex domain. Comm Math Phys, 2011, 308:641-701
[27] Morimoto Y, Wang S K, Yang T. A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation. J Math Pures Appl, 2015, 103:809-829
[28] Mouhot C, Villani C. Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch Rational Mech Anal, 2004, 173:169-212
[29] Polewczak J. Classical solutions of the Boltzmann equation in R³:asymptotic behavior of solutions. J Stat Phys, 1988, 50:61-632
[30] Ukai S, Yang T. The Boltzmann equation in the space L²∩L_β^∞:Global and time-periodic solutions. Anal Appl, 2006, 4:263-310
[31] Villani C. A review of mathematical topics in collisional kinetic theory//Handbook of mathematical Fluid Dynamics. Vol I. Amsterdam:North-Holland, 2002:71-305

MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION

MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION

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