In this talk, we study the gradient flow with respect to the Wasserstein metric of the Maximum Mean Discrepancy associated with the Coulomb kernel. In this context, we present several sufficient conditions for global convergence of the gradient flow to the unique global minimum. For instance, on closed Riemannian manifolds, we prove that the so-called Polyak-Lojasiewicz condition holds in some cases, resulting in an exponential convergence. To obtain this result, we use standard estimates from potential theory. An other result is the fact that there is no local minimum apart from the global one. This result is proven using flow interchange techniques.