A New Variational Approach to the Stability of Gravitational Systems

• Published in: Communications in mathematical physics Vol. 302; no. 1; pp. 161 - 224
• Main Authors: Lemou, Mohammed, Méhats, Florian, Raphaël, Pierre
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2011; Springer; Springer Verlag
• Subjects: Analysis of PDEs; Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Mathematical analysis; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Other topics in mathematical methods in physics; Partial differential equations; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Sciences and techniques of general use; Theoretical; Stellar System; Variational Approach; Nonlinear Stability; Monotonicity Formula; Orbital Stability; Three-dimensional calculations; Hamiltonians; Mechanical system; Steady state; Steady state solution; Mathematical physics; Numerical stability; Concentration compactness; Non linear stability; galaxies; models; nonlinear stability; stellar dynamics; orbital stability; states
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-010-1182-9

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 (MNRAS 241:15, 1989 ), (Mon. Not. R. Astr. Soc. 144:189–217, 1969 ), (Mon. Not. R. Ast. Soc. 223:623–646, 1988 ) we use the monotonicity of the Hamiltonian under generalized symmetric rearrangement transformations to prove that non increasing steady solutions are the 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.