Stable and Unstable Equilibria of Uniformly Rotating Self-Gravitating Cylinders

Abstract

The equilibrium configurations of self-gravitating uniformly rotating isothermal cylinders in contact with a heat bath and their stability is studied by recently derived analytical techniques. The known critical temperature Tc obtained by Katz and Lynden-Bell is found to be a stability threshold with respect to axially symmetric perturbations. We provide the almost explicit expression of negative specific heat solutions whose densities are sharply concentrated either near the symmetry axis or near some off-axis filaments as [Formula: see text]. The critical angular frequency observed numerically in literature is found to be the threshold value for the existence of these off-axis filaments. This is in striking contrast with the static case analyzed by Katz and Lynden-Bell where equilibrium configurations are found only if T > Tc and no negative specific heat equilibria exists at all. Metastability of the free energy's relative maximizers for T ≤ Tc is also discussed. Those off-axis configurations were predicted in the study of negative temperature states for guiding-centre plasmas and vortex systems.

Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence

“…the "first" point where the first eigenvalue vanishes) for semilinear elliptic equations, see [1], [14], usually uses the fact that the first eigenvalue is simple and that the first eigenfunction does not change sign in Ω. None of these properties is granted in our case, since the linearization of (P λ ) naturally yields first eigenfunctions which change sign [5], and first eigenvalues which need not being simple, see for example [2]. As mentioned above, we have found a natural physical assumption which allows the solution of these difficulties.…”

Supercritical Mean Field Equations on Convex Domains and the Onsager’s Statistical Description of Two-Dimensional Turbulence

“…Together with other well known physical [13], [71], [66], [72], [74], [78] and geometrical [20], [41], [75] applications, these new results were the motivation for the lot of efforts in the understanding of the resulting mean field [17], [18] Liouville-type [51] equations. We refer the reader to [3], [5], [11], [12], [15], [16], [19], [21], [22], [23], [24], [25], [26], [27], [28], [29], [32], [33], [34], [42], [46], [48], [49], [50], [52], [53], [56], [57], [60], [61], [65], [67], [69], [70], [73], [77], and more recently…”

On the global bifurcation diagram of the Gel'fand problem

“…This might seem not a good point of view since, (P λ ) being a constrained problem, the associated linearized equation is more difficult to analyze. In general the first eigenvalue σ 1,λ is not simple and the first eigenfunctions may change sign, see [3] or the Appendix below for an example of this sort. Therefore, even if we know that σ 1,λ > 0 (see [13] and [7]), it is not clear how to use this information to establish the monotonicity of µ λ .…”