  We develop a method of obtaining a hierarchy of new higher-order entropies in
the context of compressible models with local and non-local diffusion and
isentropic pressure. The local viscosity is allowed to degenerate as the
density approaches vacuum. The method provides a tool to propagate initial
regularity of classical solutions provided no vacuum has formed and serves as
an alternative to the classical energy method. We obtain a series of global
well-posedness results for state laws in previously uncovered cases including
$p(\rho) = c_p \rho$. As an application we prove global well-posedness of
collective behavior models with pressure arising from agent-based Cucker-Smale
system.
