  Non-linear versions of log-Sobolev inequalities, that link a free energy to
its dissipation along the corresponding Wasserstein gradient flow (i.e.
corresponds to Polyak-Lojasiewicz inequalities in this context), are known to
provide global exponential long-time convergence to the free energy minimizers,
and have been shown to hold in various contexts. However they cannot hold when
the free energy admits critical points which are not global minimizers, which
is for instance the case of the granular media equation in a double-well
potential with quadratic attractive interaction at low temperature. This work
addresses such cases, extending the general arguments when a log-Sobolev
inequality only holds locally and, as an example, establishing such local
inequalities for the granular media equation with quadratic interaction either
in the one-dimensional symmetric double-well case or in higher dimension in the
low temperature regime. The method provides quantitative convergence rates for
initial conditions in a Wasserstein ball around the stationary solutions. The
same analysis is carried out for the kinetic counterpart of the gradient flow,
i.e. the corresponding Vlasov-Fokker-Planck equation. The local exponential
convergence to stationary solutions for the mean-field equations, both elliptic
and kinetic, is shown to induce for the corresponding particle systems a fast
(i.e. uniform in the number or particles) decay of the particle system free
energy toward the level of the non-linear limit.
