Reduction and a Concentration-Compactness Principle for Energy-Casimir Functionals

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 [8] in a more specific setting and connect our stability analysis with the one of G. Wolansky [13].

Existence and Non-linear Stability of Rotating Star Solutions of the Compressible Euler–Poisson Equations

“…In order to prove Theorem 2.3, we will need the following lemma which is proved in [29], and uses a concentration-compactness argument.…”

“…We prove (2.16), and apply Lemma 2.8 to prove (2.14). We begin with a splitting as in [29]. For ρ ∈ W M , for any 0 < R 1 < R 2 , we have…”

A constraint variational problem arising in stellar dynamics

“…By formulating the problem in terms of spatial densities ρ = f (., v)dv in [6], the author naturally reduced it to the problem of minimizing a functional of the form…”

“…The notion of reduction and the exact relations between Q and Φ are carefully analyzed in [6], where a concentration-compactness type argument is used to deal with the variational problem. A. Burchard and Y. Guo showed that it suffices to restrict the minimization procedure to the set of symmetrically decreasing functions ρ (cf.…”

“…We consider the associated variational problem and present a new proof that puts it in the general framework for tackling the variational problems of this type, given by Y. Guo and G. Rein (cf. Rein, 2005 andRein, 2002). …”