Stable Steady States in Stellar Dynamics

We prove the existence and nonlinear stability of steady states of the Vlasov-Poisson system in the stellar dynamics case. The steady states are obtained as minimizers of an energy-Casimir functional from which fact their dynamical stability is deduced. The analysis applies to some of the well-known polytropic steady states, but it also considerably extends the class of known steady states.     

Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions

Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proved that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proved for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established.     

Stability of Spherically Symmetric Steady States in Galactic Dynamics¶against General Perturbations

Certain steady states of the Vlasov-Poisson system can be characterized as minimizers of an energy-Casimir functional, and this fact implies a non-linear stability property of such steady states. In previous investigations by Y. Guo and G. Rein, stability was obtained only with respect to spherically symmetric perturbations. In the present investigation we show how to remove this non-physical restriction.     

Global existence of classical solutions to the vlasov-poisson system in a three-dimensional,cosmological setting

The initial-value problem for the Vlasov-Poisson system is by now well understood in the case of an isolated system where, by definition, the distribution function of the particles as well as the gravitational potential vanish at spatial infinity. Here we start with homogeneous solutions, which have a spatially constant, non-zero mass density and which describe the mass distribution in a Newtonian model of the universe. These homogeneous states can be constructed explicitly, and we consider deviations from such homogeneous states, which then satisfy a modified version of the Vlasov-Poisson system. We prove global existence and uniqueness of classical solutions to the corresponding initial-value problem for initial data which represent spatially periodic deviations from homogeneous states.     

On Nonlinear Stability of Isotropic Models in Stellar Dynamics

Consider an aggregation of mass particles in space which attract each other according to Newton's law of attraction. This system can be described by distribution functions satisfying the Vlasov equation and the Poisson laws. We obtain the nonlinear stability of certain stationary states, including those obeying modified Emden's laws. A priori estimates of the energy-Casimir functions around stationary states are established for distribution functions for which the L^5/3L^{5/3}-norms of the density functions are uniformly bounded. A uniform bound on the kinetic energy of the system readily implies that these norms of the density functions are indeed uniformly bounded. In this way we prove nonlinear stability.     

Asymptotic Behaviour for the Vlasov-Poisson System in the Stellar-Dynamics Case

We study an optimal inequality which relates potential and kinetic energies in an appropriate framework for bounded solutions of the Vlasov-Poisson (VP) system. Optimal distribution functions, which are completely characterized, minimize the total energy. From this variational approach, we deduce bounds for the kinetic and potential energies in terms of conserved quantities (mass and total energy) of the solutions of the VP system and a nonlinear stability result. Then we apply our estimates to the study of the large-time asymptotics and observe two different regimes.     

Steady States in Galactic Dynamics

By minimizing the energy for the Vlasov-Poisson system under a constraint, Guo and Rein have constructed a large class of isotropic, spherically symmetric steady states. They have shown that an isolated minimizer is automatically dynamically stable under general (i.e., not necessarily symmetric) perturbations. The main result of this work is to remove the assumption that the minimizer must be isolated, so minimizers are stable even if they are not isolated. It is also shown that the Lagrange multipliers associated with all minimizers have the same value. Finally, an example where two distinct minimizers exist is studied numerically.     

Stability of Periodic Clusters in Globally Coupled Maps

The phenomenon of partial synchronization, or clustering, in a system of globally coupled C ¹-smooth maps is analyzed. We prove the stability of equally populated K-clustered states with n-periodic temporal dynamics, referred to as P _n C _K-states. For this purpose, we first obtain formulas giving a relation between longitudinal and transverse multipliers of the in-cluster periodic orbits, and then, using these formulas, we find exact parameter intervals for transverse stability. We conclude that, typically, for symmetric P _n C _K-states, in-cluster stability implies transverse stability. Moreover, transverse stability can take place even if in-cluster dynamics are unstable.     

Global stability for convection when the viscosity has a maximum

Until now, an unconditional nonlinear energy stability analysis for thermal convection according to Navier-Stokes theory had not been developed for the case in which the viscosity depends on the temperature in a quadratic manner such that the viscosity has a maximum. We analyse here a model of non-Newtonian fluid behaviour that allows us to develop an unconditional analysis directly when the quadratic viscosity relation is allowed. By unconditional, we mean that the nonlinear stability so obtained holds for arbitrarily large perturbations of the initial data. The nonlinear stability boundaries derived herein are sharp when compared with the linear instability thresholds.Received: 9 April 2003, Accepted: 28 April 2003, Published online: 12 December 2003PACS: 03.50.De, 04.20.-q, 42.65.-kCorrespondence to B. Straughan     

Über die instabilität konservativer systeme mit gyroskopischen kräften

Summary Routh's theorem states that a steady motion of a discrete, conservative mechanical system is stable if the dynamic potential W(q)=U(q)–T₀(q) assumes a minimum. This is a generalized version of the theorem on the stability of equilibrium at a minimum of the potential energy, which is due to Dirichlet. It is well known that a steady motion may also be stable if W(q) assumes a maximum instead of a minimum. The stability is then due to the gyroscopic terms in the equations of motion, without which the steady motion would be unstable. Here it is shown that the steady motion is always unstable if not only W(q) but also H ₀(q) assumes a maximum, H ₀(q) being the part of the Hamiltonian that does not depend on the momenta. It is astonishing that this unexpectedly simple criterion was not found before now. In the proof, a variational formulation is used for the problem, and the instability is shown directly from the existence of certain motions which diverge from the trivial solution.

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Vorgelegt von C. Truesdell     

A Nonlinear Stability Analysis for Thermoconvective Magnetized Ferrofluid Saturating a Porous Medium

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, and medium permeability, Da, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter (M ₃) and Darcy number (Da), the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.     

$${\mathcal{A}}$$-Stability of Global Attractors of Competition Diffusion Systems

We study the structural stability of global attractors (A{\mathcal{A}}-stability) for two-species competition diffusion systems with Morse-Smale structure. Such systems generate semiflows on positive cones of certain infinite-dimensional Banach spaces (e.g., fractional order spaces). Our main result states that a two species competition diffusion system with Morse-Smale structure is structurally A{\mathcal{A}}-stable, which implies that the set of nonlinearities for which the system possesses Morse-Smale structure is open in an appropriate space under the topology of C ²-convergence on compacta. Moreover, we provide a sufficient condition under which a system has Morse-Smale structure and provide some examples which satisfy the sufficient condition.     

Nonlinear stability of a convective motion in a porous layer driven by a horizontally periodic temperature gradient

The nonlinear global exponential pointwise stability of a vertical steady flow driven by a horizontal periodic temperature gradient in a porous layer is performed. It is shown that the stability threshold depends on the supremum of a quadratic functional, having non constant coefficients, and new in the literature on the convection problem. In solving the variational problem, a suitable functional transformation is used.Received: 27 January 2003, Accepted: 10 March 2003, Published online: 12 September 2003 Correspondence toF. Capone     

Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L^p for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.     

Simulation of shockwave propagation with a thermal lattice Boltzmann model

A two‐dimensional 19‐velocity (D2Q19) lattice Boltzmann model which satisfies the conservation laws governing the macroscopic and microscopic mass, momentum and energy with local equilibrium distribution order O(u⁴) rather than the usual O(u³) has been developed. This model is applied to simulate the reflection of shockwaves on the surface of a triangular obstacle. Good qualitative agreement between the numerical predictions and experimental measurements is obtained. As the model contains the higher‐order terms in the local equilibrium distribution, it performs much better in terms of numerical accuracy and stability than the earlier 13‐velocity models with the local equilibrium distribution accurate only up to the second order in the velocity u. Copyright © 2003 John Wiley & Sons, Ltd.     

Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in L ^p -norm for the smoothed rarefaction wave. We then employ the L ²-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity. P. Zhu was supported by JSPS postdoctoral fellowship under P99217.     

Nonlinear stability of discrete shocks for systems of conservation laws

In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the L ^p-norm for all p 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in L ^p (P 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.     

On the Influence of the Kinetic Energy¶on the Stability of Equilibria¶of Natural Lagrangian Systems

Natural Lagrangian systems (T,Π) on R ² described by the equation are considered, where is a positive definite quadratic form in and Π(q) has a critical point at 0. It is constructively proved that there exist a C ^∞ potential energy Π and two C ^∞ kinetic energies T and such that the equilibrium q(t)≡ 0 is stable for the system (T,Π) and unstable for the system . Equivalently, it is established that for C ^∞ natural systems the kinetic energy can influence the stability. In the analytic category this is not true. Accepted: October 20, 1999     

Gravitational Stability of the Interface in Water Over Steam Geothermal Reservoirs

Stability of a geothermal system is considered in a case when the water layer lies over the layer of superheated vapor in a stratum having relatively low permeability. This stratum locates between two parallel high permeable layers. Under the assumption of smallness of advective energy transfer as compared with the conductive one, the stationary distribution of the characteristics in the stratum with an interface of phase transition is obtained. The interface separates the domains occupied by water and vapor. Investigation of normal stability of the interface shows, that stable configurations in the geothermal system under consideration exist within the range of permeability values bounded by k 0.6 × 10^–15 m² from above. The most unstable configurations occur to be the quiescent states when the permeability exceeds a certain threshold. A sufficiently high value of permeability, satisfying the criterion of smallness of the advective energy transfer as compared with the conductive one makes it possible to explain the existence of a wide class of stable natural geothermal reservoirs, where the vapor layer underlies the water one.