Stability for the spherically symmetric Einstein-Vlasov system - a coercivity estimate

TL;DR

This paper establishes a coercivity estimate for the energy-Casimir functional around isotropic steady states in the spherically symmetric Einstein-Vlasov system, advancing towards a nonlinear stability proof.

Contribution

It provides a new coercivity estimate for the quadratic expansion of the energy-Casimir functional, crucial for analyzing stability of steady states in the Einstein-Vlasov system.

Findings

Quadratic part of the energy-Casimir functional is positive definite under certain conditions.

The result applies to steady states with decreasing particle energy distribution.

The estimate is a key step towards nonlinear stability analysis.

Abstract

The stability of static solutions of the spherically symmetric, asymptotically flat Einstein-Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main result is a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state. The estimate shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. This should be an essential step in a fully non-linear stability analysis for the Einstein-Vlasov system. In the present paper it is exploited for obtaining a linearized stability result.