A new variational approach to the stability of gravitational systems

TL;DR

This paper introduces a novel variational method leveraging Hamiltonian monotonicity to establish the nonlinear stability of certain gravitational stellar systems, advancing understanding beyond linear analysis.

Contribution

It develops a new variational approach based on Hamiltonian monotonicity and symmetric rearrangements to prove stability of nonincreasing steady states in gravitational systems.

Findings

Proves nonlinear stability of nonincreasing steady states under symmetric perturbations.

Introduces a variational framework inspired by physics literature.

Establishes local minimality of steady states via Hamiltonian monotonicity.

Abstract

We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the flow. This was proved at the linear level by several authors based on the pioneering work by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this paper, inspired by pioneering works from the physics litterature (Lynden-Bell 94, Wiechen-Ziegler-Schindler MNRAS 88, Aly MNRAS 89), we use the monotonicity of the Hamiltonian under generalized symmetric rearrangement transformations to prove that non increasing steady solutions are local minimizer of the Hamiltonian under equimeasurable constraints, and extract compactness from suitable minimizing sequences. This implies the nonlinear stability of nonincreasing anisotropic steady states under radially symmetric perturbations.