×

On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions. (English) Zbl 0918.35136

Summary: We are interested in the influence of grazing collisions, with deflection angle near \(\pi/2\), in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-\(s\)th-power force laws, and we deal with general initial data with bounded mass, energy, and entropy.
First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, for \(s\geq 7/3\). Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value \(\pi/2\). For very soft interactions, \(2< s<7/3\), the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term.

MSC:

35Q72 Other PDE from mechanics (MSC2000)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] L. Arkeryd, On the Boltzmann equation,Arch. Rat. Mech. Anal. 45:1–34 (1972). · Zbl 0245.76060
[2] L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,Arch. Rat. Mech. Anal. 77:11–21 (1981). · Zbl 0547.76085
[3] A. Arsenev and O. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation,Math. Sbornyk 181:4, 465–477 (1991). · Zbl 0724.35090
[4] A. Bers and J. L. Delcroix,Physique des plasmas (InterEdition/CNRS, 1994).
[5] H. Brezis,Analyse fonctionnelle (Masson, 1993).
[6] C. Cercignani,The Boltzmann equation and its applications (Springer-Verlag, 1988). · Zbl 0646.76001
[7] C. Cercignani, R. Illner, and M. Pulvirenti,The mathematical theory of dilute gases (Springer-Verlag, 1994). · Zbl 0813.76001
[8] F. Chvala and R. Pettersson, Weak solutions of the linear Boltzmann equation with very soft interactions,J. Math. Anal. and Appl. 191:360–379 (1995). · Zbl 0831.76075
[9] F. Chvala, T. Gustafsson, and R. Pettersson, On solutions to the linear Boltzmann equation with external electromagnetic force,SIAM J. Math. Anal. 24:3, 583–602 (1993). · Zbl 0771.76060
[10] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann operator in the Coulomb case,Math. Models and Meth. in the Appl. Sci. 2:2, 167–182 (1992). · Zbl 0755.35091
[11] L. Desvillettes, On the asymptotics of the Boltzmann equation when the collisions become grazing,Transp. Theory in Stat. Phys. 21:259–276 (1992). · Zbl 0769.76059
[12] L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations,Arch. Rat. Mech. Anal. 123:387–404 (1993). · Zbl 0784.76081
[13] R. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equation: Global existence and weak stability,Ann. Math. 130 :321–366 (1989). · Zbl 0698.45010
[14] R. Di Perna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation,Comm. Math. Phys. 120:1–23 (1988). · Zbl 0671.35068
[15] T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range,Arch. Rat. Mech. Anal. 82:1–12 (1983). · Zbl 0503.76091
[16] T. Goudon, Sur l’équation de Boltzmann homogène et sa relation avec l’équation de Landau-Fokker-Planck: Influence des collisions rasantes,CRAS 324 :265–270 (1997). · Zbl 0882.76079
[17] T. Goudon,Sur quelques questions relatives à la théorie cinétique des gaz et à l’équation de Boltzmann, Thèse Université Bordeaux 1 (1997).
[18] H. Grad, Asymptotic theory of the Boltzmann equation inRarefied Gas Dynamic, Third Symposium, Paris, Laurmann Ed., pp. 26–59 (Ac. Press 1963).
[19] K. Hamdache,Sur l’existence globale et le comportement asymptotique de quelques solutions de l’equation de Boltzmann, Thèse Université Paris 6 (1986).
[20] K. Hamdache, Estimations uniformes des solutions de l’équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck,CRAS 302:187–190 (1986). · Zbl 0597.76071
[21] L. Hormander,The analysis of linear pde, Vol. 1 (Springer, 1983).
[22] N. A. Krall and A. W. Trivelpiece,Principles of plasma physics (Mc Graw-Hill, 1964).
[23] E. Lifshitz and L. Pitaevski,Cinétique Physique. Coll. ”Physique Théorique,” L. Landau-E. Lifshitz (Mir, 1990).
[24] P. L. Lions, Compactness in Boltzmann equation via Fourier integral operators and applications. Part I, II and III,J. Math. Kyoto Univ. 34:2, (39) 391–461 (1994), andJ. Math. Kyoto Univ. 34 :3, 539–584 (1994). · Zbl 0831.35139
[25] R. Pettersson, Existence theorems for the linear space homogeneous transport equation,IMA J. Appl. Math. 30:81–105 (1983). · Zbl 0528.76083
[26] R. Pettersson, On solutions and higher moments for the linear Boltzmann equation with infinite-range forces,IMA J. Appl. Math. 38:151–166 (1987). · Zbl 0654.45007
[27] R. Pettersson, On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces,J. Stat. Phys. 59:1/2, 403–440 (1990). · Zbl 1083.82530
[28] M. Reed and B. Siimon,Methods of modern mathematical physics., Vol. 2 (Ac. Press, 1970). · Zbl 0197.28605
[29] S. Semmes, A primer on Hardy spaces,Comm. Part. Diff. Eqt. 19:277–319 (1994). · Zbl 0836.35030
[30] J. Simon, Compact sets in Lp(0,T;B), Ann. Mat. Pura Appl. IV:146, 65–96 (1987). · Zbl 0629.46031
[31] S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,Japan J. Appl. Math. 1:141–156 (1984). · Zbl 0597.76072
[32] C. Villani,On a new class of weak solutions to the Boltzmann and Landau equations. Personal communication (1997).
[33] B. Wennberg, On moments and uniqueness for solutions to the space homogeneous Boltzmann equation,Transp. Th. Stat. Phys. 24:4, 533–539 (1994). · Zbl 0812.76080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.