  In this paper, we study the long time behaviour of the Fokker-Planck and the
kinetic Fokker-Planck equations with many body interaction, more precisely with
interaction defined by U-statistics, whose macroscopic limits are often called
McKean-Vlasov and Vlasov-Fokker-Planck equations respectively. In the
continuity of the recent papers [63, [43],[42]] and [44, [74],[75]], we
establish nonlinear functional inequalities for the limiting McKean-Vlasov SDEs
related to our particle systems. In the first order case, our results rely on
large deviations for U-statistics and a uniform logarithmic Sobolev inequality
in the number of particles for the invariant measure of the particle system. In
the kinetic case, we first prove a uniform (in the number of particles)
exponential convergence to equilibrium for the solutions in the weighted
Sobolev space H 1 ($\mu$) with a rate of convergence which is explicitly
computable and independent of the number of particles. In a second time, we
quantitatively establish an exponential return to equilibrium in Wasserstein's
W 2 --metric for the Vlasov-Fokker-Planck equation.
