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In this paper we establish some explicit and sharp estimates of the spectralgap and the log-Sobolev constant for mean field particles system, uniform inthe number of particles, when the confinement potential have many localminimums. Our uniform log-Sobolev inequality, based on Zegarlinski's theoremfor Gibbs measures, allows us to obtain the exponential convergence in entropyof the McKean-Vlasov equation with an explicit rate constant, generalizing theresult of [10] by means of the displacement convexity approach, or [19, 20] byBakry-Emery technique or the recent [9] by dissipation of the Wassersteindistance.

Uniform Poincar{\'e} and logarithmic Sobolev inequalities for mean field particles systems