  Grad's moment models for Boltzmann equation were recently regularized to
globally hyperbolic systems, and thus the regularized models attain local
well-posedness for Cauchy data. The hyperbolic regularization is only related
to the convection term in Boltzmann equation. We in this paper studied the
regularized models with the presentation of collision terms. It is proved that
the regularized models are linear stability at the local equilibrium and
satisfy Yong's first stability condition with commonly used approximate
collision terms, and particularly with Boltzmann's binary collision model.
