  In this paper, we study the convergence to the stable equilibrium for
Kuramoto oscillators. Specifically, we derive estimates on the rate of
convergence to the global equilibrium for solutions of the Kuramoto-Sakaguchi
equation in a large coupling strength regime from generic initial data. As a
by-product, using the stability of the equation in the Wasserstein distance, we
quantify the rate at which discrete Kuramoto oscillators concentrate around the
global equilibrium. In doing this, we achieve a quantitative estimate in which
the probability that the oscillators will concentrate at the given rate tends
to one as the number of oscillators increases. Among the essential steps in our
proof are: 1) An entropy production estimate inspired by the formal Riemannian
structure of the space of probability measures, first introduced by F. Otto in
[35]; 2) A new quantitative estimate on the instability of equilibria with
antipodal oscillators based on the dynamics of norms of the solution in sets
evolving by the continuity equation; 3) The use of generalized local
logarithmic Sobolev and Talagrand type inequalities, similar to the ones
derived by F. Otto and C. Villani in [36]; 4) The study of a system of coupled
differential inequalities, by a treatment inspired by the work of L.
Desvillettes and C. Villani [13]. Since the Kuramoto-Sakaguchi equation is not
a gradient flow with respect to the Wasserstein distance, we derive such
inequalities under a suitable fibered transportation distance.
