1015 0022-4715/04/0200-1015/0 © 2004 Plenum Publishing Corporation Journal of Statistical Physics, Vol. 114, Nos. 3/4, February 2004 (© 2004) Nonlinear Functionals of Multi-D Discrete Velocity Boltzmann Equations Mikhail Feldman 1 and Seung-Yeal Ha 1 1 Department of Mathematics, University of Wisconsin-Madison; e-mail: {feldman, ha}@ math.wisc.edu Received February 4, 2003; accepted June 25, 2003 In this paper, we study nonlinear functionals measuring potential interactions and L 1 -distance between two mild solutions for the multi-dimensional discrete velocity Boltzmann equations when the initial data are a small perturbation of a vacuum. We employ Bony’s dispersion estimates to show that these functionals satisfy Lyapunov type estimates which are useful for the study of time-asymp- totics and L 1 -stability of mild solutions. KEY WORDS: Discrete velocity Boltzmann equations; L 1 -stability; nonlinear functionals. 1. INTRODUCTION The purpose of this paper is to study nonlinear functionals of the multi- dimensional discrete velocity Boltzmann equations: “ t f i (x, t)+v i · N x f i (x, t)=Q i ( f, f )(x, t), (x, t) ¥ R n × R + , 1 [ i [ N, (1.1) where f i is the density of particles with velocity v i =(v 1 i ,..., v n i ) and the system is assumed to be strictly hyperbolic in the sense that all characteris- tic velocities are distinct. Moreover, we assume that the collision operator Q i ( f, f ) satisfies the transversality assumption: Q i ( f, f ) — C 1 [ j, k [ N B jk i f j f k , B jk i =0, if j=k and 0< max i, j, k |B jk i |=: B g < .. (1.2)