PROBABILITY METRICS AND UNIQUENESS OF THE SOLUTION TO THE BOLTZMANN EQUATION FOR A

PROBABILITY METRICS AND UNIQUENESS OF THE SOLUTION TO THE BOLTZMANN EQUATION FOR A MAXWELL GAS G. TOSCANI AND C. VILLANI Abstract. We consider a metric for probability densities with finite variance on Rd, and compare it with other metrics. We use it for several applications, including a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules. Key-words : spatially homogeneous Boltzmann equation, probability metrics, Maxwellian molecules. 1. Introduction Denote by Ps(Rd), s > 0, the class of all probability distributions F on Rd, d ≥ 1, such that ∫ R d |v|s dF (v) < ∞. We introduce a metric on Ps(Rd) by (1) ds(F,G) = sup ??Rd |f?(?)? g?(?)| |?|s where f? is the Fourier transform of F , f?(?) = ∫ R d e?i?·v dF (v). Let us write s = m + ?, where m is an integer and 0 ≤ ? < 1. In order that ds(F,G) be finite, it suffices that F and G have the same moments up to order m. The norm (1) has been introduced in [6] to investigate the trend to equilibrium of the solutions to the Boltzmann equation for Maxwell molecules.

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