A New Variational Approach to the Stability of Gravitational Systems

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 nonl...

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Veröffentlicht in:Communications in mathematical physics Jg. 302; H. 1; S. 161 - 224
Hauptverfasser: Lemou, Mohammed, Méhats, Florian, Raphaël, Pierre
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer-Verlag 01.02.2011
Springer
Springer Verlag
Schlagworte:
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
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Zusammenfassung: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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-010-1182-9