  We study the convergence rate to equilibrium for a family of Markov
semigroups $\{\mathcal{P}_t^{\epsilon}\}_{\epsilon > 0}$ generated by a class
of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including
Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96,
and the shell model SABRA. In the regime of vanishing, balanced noise and
dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on
the exponential convergence in terms of the small parameter $\epsilon$. By
scaling, this regime implies corresponding optimal results both for fixed
dissipation and large noise limits or fixed noise and vanishing dissipation
limits. As part of the proof, and of independent interest, we obtain
uniform-in-$\epsilon$ upper and lower bounds on the density of the stationary
measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower
bounds by a de Giorgi-type iteration (both uniform in $\epsilon$). The spectral
gap estimate on the semigroup is obtained by a weak Poincar\'e inequality
argument combined with quantitative hypoelliptic regularization of the
time-dependent problem.
