  In this paper, we establish the entropy-entropy production estimate for the
ES-BGK model, a generalized version of the BGK model of the Boltzmann equation
introduced for better approximation in the Navier-Stokes limit. Our result
improves the previous entropy production estimate [39] in that (1) the full
range of Prandtl parameters $-1/2\leq\nu <1$ including the critical case
$\nu=-1/2$ is covered, and (2) a sharper entropy production bound is obtained.
An explicit characterization of the coefficient of the entropy-entropy
production estimate is also presented.
