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.     

A stability result for the relativistic Vlasov-Maxwell system

We consider a space-periodic version of the relativistic Vlasov-Maxwell system describing a collisionless plasma consisting of electrons and positively charged ions. As our main result, we prove that certain spacially homogeneous stationary solutions are nonlinearly stable. To this end we also establish global existence of weak solutions to the corresponding initial value problem. Our investigation is motivated by a corresponding result for the Vlasov-Poisson system, cf. [1, 14].     

Non-Linear Stability of Gaseous Stars

We construct steady states of the Euler-Poisson system with a barotropic equation of state as minimizers of a suitably defined energy functional. Their minimizing property implies the non-linear stability of such states against general, i.e., not necessarily spherically symmetric, perturbations. The mathematical approach is based on previous stability results for the Vlasov-Poisson system by Y. Guo and G. Rein, exploiting the energy-Casimir technique. The analysis is conditional in the sense that it assumes the existence of solutions to the initial value problem for the Euler-Poisson system which preserve mass and energy. The relation between the Euler-Poisson and the Vlasov-Poisson system in this context is also explored. (Accepted January 30, 2003) Published online May 14, 2003 Communicated by P.J. Holmes     

Gas Flows with Several Thermal Nonequilibrium Modes

We study gas flows with any finite number of thermal nonequilibrium modes. The equations describing such flows are a hyperbolic system with several relaxation equations. An important feature is entropy increase dictated by physics for any irreversible process. Under physical assumptions we obtain properties of thermodynamic variables relevant to stability. By the energy method we prove global existence and uniqueness for the Cauchy problem when the initial data are small perturbations of constant equilibrium states. We give a precise formulation of the fundamental solution for the linearized system, and use it to obtain large time behavior of solutions to the nonlinear system. In particular, we show that the entropy increases but stays bounded. The resulting asymptotic picture of nonequilibrium flows is in a pointwise sense both in space and in time.     

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.     

Buckling and nonlinear dynamics of elastically coupled double-beam systems

This paper deals with damped transverse vibrations of elastically coupled double-beam system under even compressive axial loading. Each beam is assumed to be elastic, extensible and supported at the ends. The related stationary problem is proved to admit both unimodal (only one eigenfunction is involved) and bimodal (two eigenfunctions are involved) buckled solutions, and their number depends on structural parameters and applied axial loads. The occurrence of a so complex structure of the steady states motivates a global analysis of the longtime dynamics. In this regard, we are able to prove the existence of a global regular attractor of solutions. When a finite set of stationary solutions occurs, it consists of the unstable manifolds connecting them.     

A variational approach for materially stable anisotropic hyperelasticity

In this work we propose an anisotropic stored energy function which satisfies a priori the Legendre–Hadamard condition, which is strongly related to the material stability of the constitutive equations. In the linearized case this condition implies positive wave speeds. The Legendre–Hadamard condition plays also an important role for the (local) existence of solutions in the neighborhood of stationary points. We apply the proposed hyperelastic energies to soft tissues and compare the formulation with existing models which have been used for the calculation of medial collateral ligament and arterial walls. In our numerical and analytical investigations we discuss the distribution of wave speeds for a sequence of deformation states containing some essential stress–strain characteristics of the compared models.     

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.     

Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation

We study the global-in-time behavior of solutions to a reaction–diffusion system with mass conservation, as proposed in the study of cell polarity, particularly, the second model of the work by Otsuji et al. (PLoS Comput Biol 3:e108, 2007). First, we show the existence of a Lyapunov function and confirm the global-in-time existence of the solution with compact orbit. Then we study the stability and instability of stationary solutions by using the semi-unfolding-minimality property and the spectral comparison. As a result the dynamics near the stationary solutions is qualitatively characterized by a variational function.     

A Nonlinear Model of Age-Dependent Population Dynamics

We consider a mathematical model of age-dependent population dynamics that is a generalization of the Gurtin–MacCamy model. We study the existence and uniqueness of solutions of an initial boundary-value problem and the existence and stability of stationary age distributions.     

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.     

Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit

We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.     

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔,分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布;证明了Ｈｏｐｆ分岔的存在性,通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性,讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

LINEAR AND NONLINEAR DYNAMICS OF CANTILEVERED CYLINDERS IN AXIAL FLOW. PART 1: PHYSICAL DYNAMICS

This paper is the first in a three-part study of the dynamics of cantilevered cylinders in axial flow. After an extensive literature review, the physical dynamics of the system is examined; specifically (a) the experimental behaviour of elastomer cylinders in water flow, and (b) the energy transfer mechanisms, discussed from a work–energy perspective without solving the equations of motion. In general, the system loses stability by divergence and, as the flow velocity is increased, it is subject to second- and third-mode flutter, provided that the free end is well-streamlined; if, however, the free end is blunt, these instabilities do not occur. Oscillations are generally three-dimensional (orbital). The experimental observations are in good qualitative agreement with those expected from the energy transfer analysis, and in reasonably good quantitative agreement with solutions of the linearized equation of motion (obtained from Part 3 of this study). For some shapes of the free-end, resonances are observed with a fairly constant Strouhal number.     

Asymptotic Stability of a Stationary Solution to a Thermal Hydrodynamic Model for Semiconductors

The present paper concerns the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a one-dimensional heat-conductive hydrodynamic model for semiconductors. It is important to analyze thermal influence on the motion of electrons in semiconductor device to improve the reliability of devices by handling a hot carrier problem. We show the unique existence of the stationary solution satisfying a subsonic condition by using the Leray–Schauder and the Schauder fixed-point theorems. Then the asymptotic stability of the stationary solution is proved by deriving the a priori estimate uniformly in time. Here an energy form plays an essential role. We also prove that the solution converges to the stationary solution exponentially fast as time tends to infinity.     

Spatiotemporal Patterns of a Homogeneous Diffusive Predator–Prey System with Holling Type III Functional Response

The dynamics of a diffusive predator–prey system with Holling type-III functional response subject to Neumann boundary conditions is investigated. The parameter region for the stability and instability of the unique constant steady state solution is derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method and Leray–Schauder degree theory. The effect of various parameters on the existence and nonexistence of spatiotemporal patterns is analyzed. These results show that the impact of Holling type-III response essentially increases the system spatiotemporal complexity.     

On a class of quasilinear Schrodinger equations

A type of quasilinear Schrodinger equations in two space dimensions which describe attractive Bose-Einstein condensates in physics is discussed. By establishing the property of the equation and applying the energy method, the blowup of solutions to the equation are proved under certain conditions. At the same time, by the variational method, a sutficient condition of global existence which is related to the ground state of a classical elliptic equation is obtained.