On Krein-like theorems for noncanonical Hamiltonian systems with continuous spectra: application to Vlasov-Poisson

The notions of spectral stability and the spectrum for the Vlasov-Poisson system linearized about homogeneous equilibria, f_0(v), are reviewed. Structural stability is reviewed and applied to perturbations of the linearized Vlasov operator through perturbations of f_0. We prove that for each f_0 there is an arbitrarily small delta f_0' in W^{1,1}(R) such that f_0+delta f_0$is unstable. When$f_0$ is perturbed by an area preserving rearrangement, f_0 will always be stable if the continuous spectrum is only of positive signature, where the signature of the continuous spectrum is defined as in previous work. If there is a signature change, then there is a rearrangement of f_0 that is unstable and arbitrarily close to f_0 with f_0' in W^{1,1}. This result is analogous to Krein's theorem for the continuous spectrum. We prove that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in C^n norm that makes f_0 unstable. If f_0 is stable we prove that the signature of every discrete mode is the opposite of the continuum surrounding it.Comment: Submitted to the journal Transport Theory and Statistical Physics. 36 pages, 12 figure

Orbital Stability of Multi-Planet Systems: Behavior at High Masses

In the coming years, high contrast imaging surveys are expected to reveal the characteristics of the population of wide-orbit, massive, exoplanets. To date, a handful of wide planetary mass companions are known, but only one such multi-planet system has been discovered: HR8799. For low mass planetary systems, multi-planet interactions play an important role in setting system architecture. In this paper, we explore the stability of these high mass, multi-planet systems. While empirical relationships exist that predict how system stability scales with planet spacing at low masses, we show that extrapolating to super-Jupiter masses can lead to up to an order of magnitude overestimate of stability for massive, tightly packed systems. We show that at both low and high planet masses, overlapping mean motion resonances trigger chaotic orbital evolution, which leads to system instability. We attribute some of the difference in behavior as a function of mass to the increasing importance of second order resonances at high planet-star mass ratios. We use our tailored high mass planet results to estimate the maximum number of planets that might reside in double component debris disk systems, whose gaps may indicate the presence of massive bodies.Comment: Accepted to Ap

Efficient solutions of two-dimensional incompressible steady viscous flows

A simple, efficient, and robust numerical technique is provided for solving two dimensional incompressible steady viscous flows at moderate to high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established