Reduction and a concentration-compactness principle for energy-Casimir functionals

TL;DR

This paper develops a reduction technique for energy-Casimir functionals, establishing a concentration-compactness principle for mass densities, which aids in analyzing the existence and stability of steady states in conservative systems.

Contribution

It introduces reduced functionals acting on space densities, generalizes previous methods, and connects stability analysis with Lions' concentration-compactness principle.

Findings

Established compactness properties for reduced functionals

Proved existence of minimizers for energy-Casimir functionals

Linked stability analysis with Lions' concentration-compactness principle

Abstract

Energy-Casimir functionals are a useful tool for the construction of steady states and the analysis of their nonlinear stability properties for a variety of conservative systems in mathematical physics. Recently, Y. Guo and the author employed them to construct stable steady states for the Vlasov-Poisson system in stellar dynamics, where the energy-Casimir functionals act on number density functions on phase space. In the present paper we construct natural, reduced functionals which act on mass densities on space and study compactness properties and the existence of minimizers in this context. This puts the techniques developed by Y.Guo and the author into a more general framework. We recover the concentration-compactness principle due to P.L.Lions in a more specific setting and connect our stability analysis with the one of G.Wolansky.