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

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
CeΛbt , Λb 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.