Monday, June 25, 2012

1206.5147 (Emanuele Dolera et al.)

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

Emanuele Dolera, Eugenio Regazzini
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 C e^{Lambda_b t}, Lambda_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 Lambda_b, to the cost of the "explosion" of the constant 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.
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