Entropy Hierarchies for Equations of Compressible Fluids and Self-Organized Dynamics

Abstract

We develop a method of obtaining a hierarchy of new higher-order entropies in the context of compressible models with local and nonlocal 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 an agent-based Cucker--Smale system.