  In this paper, we investigate gradient estimate of the Poisson equation and
the exponential convergence in the Wasserstein metric $W_{1,d_{l^1}}$, uniform
in the number of particles, and uniform-in-time propagation of chaos for the
mean-field weakly interacting particle system related to McKean-Vlasov
equation. By means of the known approximate componentwise reflection coupling
and with the help of some new cost function, we obtain explicit estimates for
those three problems, avoiding the technical conditions in the known results.
Our results apply when the confinement potential $V$ has many wells, the
interaction potential $W$ has bounded second mixed derivative $\nabla^2_{xy}W$
which should be not too big so that there is no phase transition. As an
application, we obtain the concentration inequality of the mean-field
interacting particle system with explicit and sharp constants, uniform in time.
Several examples are provided to illustrate the results.
