On a Priori Estimates of Nonlinear Stability in Spherically Symmetric Stellar Dynamics.

Details On a Priori Estimates of Nonlinear Stability in Spherically Symmetric Stellar Dynamics. On a Priori Estimates of Nonlinear Stability in Spherically Symmetric Stellar Dynamics.

Jan 1990

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1990phdt........91s&link_type=abstract

Thesis (PH.D.)--STATE UNIVERSITY OF NEW YORK AT BUFFALO, 1990.Source: Dissertation Abstracts International, Volume: 51-08, Sect

Mathematics

Symplectic Geometry

Scientific paper

There are a number of reasons to consider Hamiltonian structures of mechanical systems. These structures have had considerable use in the study of stability. Recent advances in symplectic geometry, combined with the Liapunov technique developed by Arnold, have set the stage for new applications in nonlinear stability analysis. Arnold's method only works under smoothness assumptions on an equilibrium solution; nevertheless, generalizations of his idea to nonsmooth cases have been carried out for vortex patches and some planar Euler flows. The aim of this work is to extend the generalizations to stellar dynamics. The stellar dynamical system can be described by distribution functions f(t, x, v), (t, x, v) in IR times IR^3 times IR^3, satisfying the coupled Vlasov and Poisson equations. This system admits a Hamiltonian structure and has drawn great attention from mathematicians and astrophysicists because of its still-open global existence and nonlinear stability problems. The total energy plus a suitably chosen conserved quantity is used as the Lyapunov function in this analysis. Motivated by the positivity of a linear stability criterion in the astronomical literature, we found an a priori estimate of this Lyapunov function for the spherically symmetric stationary solutions which are decreasingly dependent on their energy. The other positivity property found in the literature is interpreted in our geometrical setting as the positive definiteness on the tangent space of the coadjoint orbit through a regular steady state solution. This paper enlarges the class of perturbed distribution functions in which the a priori estimate is valid. By choosing a suitable conserved quantity, we show that our stability result is also true for some general stationary solutions in a spherically symmetric stellar system. These solutions depend on both their energy and the square of angular momentum.

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