On the relation between rates of relaxation and convergence of wild sums for solutions of the Kac equation

Abstract

In the case of Maxwellian molecules, the Wild summation formula gives an expression for the solution of the spatially homogeneous Boltzmann equation in terms of its initial data F as a sum f ( v , t ) = ∑ n = 1 ∞ e - t ( 1 - e - t ) n - 1 Q n + ( F ) ( v ) . Here, Q n + ( F ) is an average over n-fold iterated Wild convolutions of F. If M denotes the Maxwellian equilibrium corresponding to F, then it is of interest to determine bounds on the rate at which ∥ Q n + ( F ) - M ∥ L 1 ( R ) tends to zero. In the case of the Kac model, we prove that for every ε > 0 , if F has moments of every order and finite Fisher information, there is a constant C so that for all n, ∥ Q n + ( F ) - M ∥ L 1 ( R ) ⩽ Cn Λ + ε where Λ is the least negative eigenvalue for the linearized collision operator. We show that Λ is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of f ( · , t ) to M. A key role in the analysis is played by a decomposition of Q n + ( F ) into a smooth part and a small part. This depends in an essential way on a probabilistic construction of McKean. It allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regularity of the initial data.