Entropy Hierarchies for equations of compressible fluids and self-organized dynamics
arXiv:1908.01784
The paper introduces a hierarchy of higher‑order entropy functionals for compressible fluid equations with possibly degenerate viscosity, providing an alternative to the classical energy method and yielding new global well‑posedness results, including for pressure laws and collective dynamics models like Cucker‑Smale.
Abstract
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 $p(Ï) = c_p Ï$. As an application we prove global well-posedness of collective behavior models with pressure arising from agent-based Cucker-Smale system.
16 pages