Existence of a minimizer to the particle number-Casimir functional for the Einstein-Vlasov system
Håkan Andréasson, Markus Kunze
TL;DR
The paper develops a rigorous variational framework for the Einstein-Vlasov system by minimizing the particle number-Casimir functional $\mathcal{D}(f)$ under a mass constraint and a density bound. A novel velocity-space rearrangement is introduced to overcome noncompactness, yielding a limit $f_0$ that remains within the admissible set and satisfies key tightness properties; the Casimir-binding-energy condition (CBEC) is shown to hold for $k\in(0,2]$, guaranteeing the existence of a minimizer $f_0$ with ${\cal D}(f_0)=I$. The analysis yields a variational inequality that implies compact support in velocity and, under favorable conditions, a static-solution form for $f_0$, with strong evidence that small-mass minimizers correspond to static Einstein-Vlasov states. A stability conjecture is proposed: isotropic static minimizers with positive Casimir-binding energy are stable under mass-preserving perturbations, linking variational structure to nonlinear stability in the relativistic regime. The work provides new compactness tools and a solid variational basis for studying static solutions and their stability in the Einstein-Vlasov setting.
Abstract
In 2001 Wolansky \cite{Wol} introduced a particle number-Casimir functional for the Einstein-Vlasov system. Two open questions are associated with this functional. First, a meaningful variational problem should be formulated and the existence of a minimizer to this problem should be established. The second issue is to show that a minimizer, for some choice of the parameters, is a static solution of the Einstein-Vlasov system. In the present work we solve the first problem by proving the existence of a minimizer to the particle number-Casimir functional. On the technical side, it is a main achievement that we are able to bypass the non-compactness of minimizing sequences by new arguments in both $v$-space and $x$-space, which might have several further applications. We note that such compactness results for the Einstein-Vlasov system have been absent in the literature, whereas similar results have been known in the Newtonian case. We also provide arguments which give strong support that minimizers corresponding to small masses are static solutions of the Einstein-Vlasov system. Furthermore, our analysis leads us to propose a new stability criterion for static solutions: We conjecture that a static solution for which the Casimir-binding energy is positive is stable for mass-preserving perturbations.