Equivalence between $N$-particle log-Sobolev inequalities and non-linear Łojasiewicz inequalities for a class of mean field systems

The convergence rate of a free energy Wasserstein gradient flow is quantified by its so-called Polyak-Lojasiewicz (PL) constant $λ$, which relates the objective function to its dissipation along the flow. Such a flow is the mean-field limit as $N$ goes to infinity of a system of $N$ interacting part