Uniform poincare and logarithmic sobolev inequalities for mean field particle systems

In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincare inequality is based on the work of Ledoux (In Seminaire de Probabilites, XXXV (2001) 167194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski's theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015-4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKeanVlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971-1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109132; Ann. Appl. Probab. 13 (2003) 540-560) (by Bakry-Emery's technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429-445) (by dissipation of the Wasserstein distance).