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. Under really mild conditions on the initial datum and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 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 \(C e^{\varLambda _b t}\), \(\varLambda _b\) being the least negative eigenvalue of the linearized collision operator and \(C\) a constant depending only on the initial datum. The validity of this quantification was conjectured, about fifty years ago, by Henry P. McKean. 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.