Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

Abstract

The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. The beginning of the story dates back to a pioneering work by Hilbert, who first formalized the concept of linearization of the collision operator and pointed out the importance of its eigenvalues with respect to a certain asymptotic behavior of the Boltzmann equation. Under really mild conditions on initial data -close to being necessary -and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1.1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form Ce Λ b t , Λ b being the least negative of the aforesaid eigenvalues and C a constant which depends only on a few simple numerical characteristics (e.g. moments) of the initial datum. The validity of this quantification was conjectured, about fifty years ago, in a paper by Henry P. McKean but, in spite of several attempts, the best answer known up to now consists in a bound with a rate which can be made arbitrarily close to Λ b , to the cost of the "explosion" of the constant * Supported in part by MIUR-2008MK3AFZ † Also affiliated with CNR-IMATI, Milano, Italy.C. Moreover, its deduction is subject to restrictive hypotheses on the initial datum, besides the Grad angular cutoff condition. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean. Our work represents in fact a confirmation of this analogy, since the techniques we develop here crucially rely on certain formulations of the central limit theorem. The proof of Theorem 1.1 starts by assuming the Grad angular cutoff and proceeds with these steps: 1) A new representation, in Theorem 1.2, for the solution of the Boltzmann equation as expectation of a random probability distribution of a weighted random sum of independent and identically distributed random vectors. 2) An upper bound for the total variation distance of interest expressed as sum of expectations of the total variation distance, between the aforesaid random probability distribution and the limiting Maxwellian law, over two appropriate events U and U c . 3) The proof that the probability of U approaches zero, as time goes to infinity, at an exponential rate equal to Λ b . 4) An extension of a classical Beurling inequality which, combined with newBerry-Esseen-like inequalities, leads to the validity of the desired exponential rate Λ b of decay also for the expectation over U c . Then, the conclusion can be extended to the case of weak cutoff hypothesis by a standard truncation argument. To complete this description of the paper, we mention the use of the aforesaid representation to characterize, in Theorem 1.3, the domain of attraction of the Maxwellian limit.Mathematics subj...

Infinite energy solutions to inelastic homogeneous Boltzmann equations

“…The existence of such a representation (5) is by no means obvious. A similar expression has been given for the collision operator modelling fully elastic Maxwell molecules recently [27]. For inelastic molecules, it is proven in Proposition 7 below.…”

“…We can now formulate the above mentioned probabilistic representation. The first representation of this type has been derived in [27] for the fully elastic Boltzmann equation in R 3 .…”

“…The extension to dimension d > 1 is by no means straightforward. Only in the very recent paper [27], Dolera and Regazzini derived a suitable probabilistic representation of the solution of the homogeneous Boltzmann equation in dimension d = 3, using particular coordinates on R 3 and its rotation group. Here, we extend the Dolera-Regazzini probabilistic representation to equation (4) with kernels of the form (5), in arbitrary dimensions d ≥ 3.…”

Characterization of Weak Convergence of Probability-Valued Solutions of General One-Dimensional Kinetic Equations

“…(See, e.g., Theorem 6.7 in [8].) It is worth recalling such a representation for the sake of expository clarity, even if analogous descriptions have already been given in [6,9,18,19,24]. Before proceeding to apply the Skorokhod theorem, it is useful to introduce some slight changes to the presentation in Subsection 2.1.…”

“…'s, for short). Furthermore, remarkable efforts have been made to discover the rate of approach to equilibrium, exactly as in allied works on the classical kinetic equations such as [11,13,16,17,21,22,23,28,29] for onedimensional models, and [12,14,18] for multidimensional ones. New problems arise in connection with the following natural questions: Can one formulate necessary conditions on the initial data in order that they produce relaxation to equilibrium?…”