##
**Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation.**
*(English)*
Zbl 1081.82616

Summary: This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order \(2+epsilon\), then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then \(\int |v|>R ^{f(v, t)} |v|^2 dv \to 0\) as \(R\to \infty\), and this convergence is uniform in time.

### MSC:

82C40 | Kinetic theory of gases in time-dependent statistical mechanics |

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

### Keywords:

Boltzmann equation; Fourier transform; probability measures; weak convergence; Prokhorov metric; bivariate distributions with given marginals; Tanaka functional
PDFBibTeX
XMLCite

\textit{E. Gabetta} et al., J. Stat. Phys. 81, No. 5--6, 901--934 (1995; Zbl 1081.82616)

Full Text:
DOI

### References:

[1] | L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,Arch. Rat. Mech. Anal. 77:11–21 (1981). · Zbl 0547.76085 |

[2] | L. Arkeryd, Stability inL 1 for the spatially homogeneous Boltzmann equation,Arch. Rat. Mech. Anal. 103:151–167 (1988). · Zbl 0654.76074 |

[3] | L. Arkeryd, Infinite range forces and strongL 1-asymptotics for the space-homogeneous Boltzmann equation, inAdvances in Analysis, Probability and Mathematical Physics, S. Albeverio, W. A. J. Luxemburg, and M. Wolff, eds. (Kluwer, Dortrecht, 1994). |

[4] | N. H. Bingham, C. M. Goldie, and J. M. Teugels, Regular variation, inEncyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1987). · Zbl 0617.26001 |

[5] | A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules,Dokl. Akad. Nauk SSSR 225:1041–1044 (1975) [in Russian]. · Zbl 0361.76077 |

[6] | A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules,Sov. Sci. Rev. c. 7:111–233 (1988). · Zbl 0850.76619 |

[7] | E. Carlen, Personal Communication. |

[8] | E. Carlen and M. Carvalho, Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation.J. Stat. Phys. 67(3/4):575–608 (1992). · Zbl 0899.76317 |

[9] | E. A. Carlen and M. C. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels,J. Stat. Phys. 74(3/4):743–782 (1994). · Zbl 0831.76074 |

[10] | E. Carlen and A. Soffer, Entropy production by block summation and central limit theorems,Commun. Math. Phys. 140:339–371 (1991). · Zbl 0734.60024 |

[11] | C. Cercignani,The Theory and Application of the Boltzmann Equation (Springer, New York, 1988). · Zbl 0646.76001 |

[12] | V. Comincioli and G. Toscani, Operator splitting of the Boltzmann equation of a Maxwell gas, inBoundary Value Problems for Partial Differential Equations and Applications, C. Baiocchi and J. L. Lions, eds. (Masson, Paris, 1993). · Zbl 0799.35208 |

[13] | L. Desvillettes, On the regularizing properties of the non cut-off Kac equation, Preprint (1994). · Zbl 0827.76081 |

[14] | R. M. Dudley, Convergence of Baire measures,Studia Math. 27:251–268 (1966). · Zbl 0147.31301 |

[15] | R. M. Dudley, Distances of probability measures and random variables,Ann. Math. Stat. 39:1563–1572 (1968). · Zbl 0169.20602 |

[16] | T. Elmroth, Gloal 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 |

[17] | M. Fréchet, Sur les tableaux de corrélation dont les marges sont données.Ann. Univ. Lyon A 14:53–77 (1951). |

[18] | E. Gabetta and G. Toscani, On convergence to equilibrium for Kac’s caricature of a Maxwellian gas,J. Math. Phys. 35:1 (1994). · Zbl 0801.76082 |

[19] | E. Gabetta and L. Pareschi, About the non cut-off Kac equation: Uniqueness and asymptotic behaviour, Preprint 911/94, Istituto di Analisi Numerica del CNR, Pavia (1994). · Zbl 0873.45006 |

[20] | E. Gabetta and L. Pareschi, The Maxwell gas and its Fourier transform, towards a numerical approximation, to appear. · Zbl 0763.76070 |

[21] | E. Gabetta, On a conjecture of McKean with applications to Kac modelTransport Theory Stat. Phys. 24: in press. |

[22] | W. Hoeffding, Masstabinvariante Korrelationstheorie,Schriften Math. Inst. Inst. Angew. Math. Univ. Berlin 5:179–233 (1940). |

[23] | L. Hörmander,The Analysis of Linear Partial Differential Operators I (Springer-Verlag, New York, 1983). |

[24] | E. Ikenberry and C. Truesdell, On the pressure and the flux of energy according to Maxwell’s kinetic energy I,J. Rat. Mech. Anal. 5:1–54 (1956). · Zbl 0070.23504 |

[25] | M. Kac,Probability and Related Topics in the Physical Sciences (New York, 1959). · Zbl 0087.33003 |

[26] | H. P. McKean, Jr., Speed of approch to equilibrium for Kac’s caricature of a Maxwellian gas,Arch. Rat. Mech. Anal. 21:343–367 (1966). · Zbl 1302.60049 |

[27] | H. Murata and H. Tanaka, An inequality for certain functional of multidimensional probability distributions,Hiroshima Math. J. 4:75–81 (1974). · Zbl 0287.60021 |

[28] | Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory.,Theor. Prob. Appl. 1:157–214 (1956). · Zbl 0075.29001 |

[29] | A. Pulvirenti and G. Toscani, On the theory of the spatially uniform Boltzmann equation for Maxwell molecules in Fourier transform,Ann. Mat. Pura Appl., in press. · Zbl 0874.45005 |

[30] | A. Pulvirenti, Alcuni risultati sull’equazione di Boltzmann, Thesis, Pavia (1994). |

[31] | A. Pulvirenti and B. Wennberg, Pointwise lower bounds for the solution of the Boltzmann equation, Preprint (1994). · Zbl 0874.45006 |

[32] | E. Ringeisen, Contributions à l’Étude Mathématique des Équations Cinétiques Ph.D. Thesis, Université Paris 7, Paris (1991). |

[33] | V. Strassen, The existence of probability measures with given marginals,Ann. Math. Stat. 36:423–439 (1965). · Zbl 0135.18701 |

[34] | H. Tanaka, An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas.Wahrsch. Verw. Geb. 27:47–52 (1973). · Zbl 0302.60005 |

[35] | H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,Wahrsch. Verw. Geb. 46:67–105 (1978). · Zbl 0389.60079 |

[36] | G. Toscani, Bivariate distributions with given marginals and applications to kinetic theory of gases, inAtti Convegno Nazionale AIMET A di Meccanica Stocastica (Taormina, 1993). |

[37] | V. S. Varadarajan, Measures on Topological Spaces,Trans. Am. Math. Soc. Ser. 2 48:161–228 (1961). |

[38] | B. Wennberg, Stability and exponential convergence for the Boltzmann equation, Thesis, Chalmers University of Technology (1993). · Zbl 0786.76074 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.