On the Stability of Stationary Solutions of the Twice-Averaged Hill’s Problem Taking into Account the Planet Oblateness

1999 ◽  
pp. 457-457
Author(s):  
M. A. Vashkovyak
Keyword(s):  
Stationary Solutions ◽  
Hill’S Problem ◽  
Hill's Problem ◽  
The Stability

scholarly journals On The Stability of Stationary Solutions of the Twice-Averaged Hill’s Problem Taking Into Account The Planet Oblateness

1999 ◽  
Vol 172 ◽  
pp. 457-457
Author(s):  
M.A. Vashkovyak
Keyword(s):  
Orbital Evolution ◽  
Stationary Solutions ◽  
Circular Orbits ◽  
Hill’S Problem ◽  
Hill's Problem ◽  
The Stability

The problem of satellite orbital evolution with the combined influence of a distant perturbing body and the planet oblateness is well known (Laplace, 1805; Lidov, 1962, 1973; Kozai, 1963; Kudielka, 1994, 1997). The case of near-circular orbits is investigated in more details in (Sekiguchi, 1961; Allan and Cook, 1964; Vashkovyak, 1974).

scholarly journals Lr-LpStability of the Incompressible Flows with Nonzero Far-Field Velocity

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Jaiok Roh
Keyword(s):  
Constant Velocity ◽  
Incompressible Flows ◽  
Temporal Stability ◽  
Stationary Solutions ◽  
Decay Rates ◽  
Navier Stokes ◽  
Far Field ◽  
Spatial Direction ◽  
Field Velocity ◽  
The Stability

We consider the stability of stationary solutionswfor the exterior Navier-Stokes flows with a nonzero constant velocityu∞at infinity. Foru∞=0with nonzero stationary solutionw, Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability inLpspaces for1<pand obtained good stability decay rates. For the spatial direction, we recently obtained some results. Foru∞≠0, Heywood (1970, 1972) and Masuda (1975) have studied the temporal stability inL2space. Shibata (1999) and Enomoto and Shibata (2005) have studied the temporal stability inLpspaces forp≥3. Then, Bae and Roh recently improved Enomoto and Shibata's results in some sense. In this paper, we improve Bae and Roh's result in the spacesLpforp>1and obtainLr-Lpstability as Kozono and Ogawa and Borchers and Miyakawa obtained foru∞=0.

scholarly journals Stability of a Hot Smoluchowski Fluid

2001 ◽  
Vol 08 (01) ◽  
pp. 19-27 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Numerical Range ◽  
Nonlinear Parabolic Equations ◽  
Stationary Solutions ◽  
Small Perturbations ◽  
Material Density ◽  
Nonlinear Parabolic ◽  
The Stability ◽  
Special Case

We study coupled nonlinear parabolic equations for a fluid described by a material density ρ and a temperature Θ, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearized equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for Θ and no-flow conditions for the material. The spectrum of the generator L of time evolution, regarded as an operator on L2[0,1], is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is ℂ.

Nonlinear convection in high vertical channels

1984 ◽  
Vol 143 ◽  
pp. 223-242 ◽  
Author(s):  
C. Normand
Keyword(s):  
Specific Treatment ◽  
Finite Size ◽  
Stationary Solutions ◽  
Amplitude Equation ◽  
Weakly Nonlinear ◽  
Convective Systems ◽  
Horizontal Extent ◽  
Aspect Ratios ◽  
Infinite Height ◽  
The Stability

Application of Landau's ideas to the theory of weakly nonlinear instabilities shows that the amplitude of the unstable modes behaves as the square root of the reduced control parameter ε, its critical value being ε = 0. When applied to cellular structures the theory has been improved by taking into account the slow spatial variations of the amplitude and phase of the unstable modes. Until now the case of thermo-convective instabilities in high vertical channels has not been studied using this approach. In high vertical structures the nonlinear terms disappear in the limit of an infinite height, and the supercritical behaviour requires a specific treatment. It differs from the standard analysis valid for the case of fluid layers of infinite horizontal extent, where the nonlinearities and the finite-size effects are disconnected. In the limit of high aspect ratios (height [Gt ] horizontal extent) we have derived an amplitude equation for convective systems where the nonlinear terms contain derivatives at the lowest order. As a consequence the amplitude equation cannot be put into a variational form and the stability of the stationary solutions cannot be deduced from an ordering in decreasing values of a Lyapunov functional.

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