Classical Solutions and the Glassey-Strauss Theorem for the 3D Vlasov-Maxwell System

R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59–90] that C ¹ solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof.     

A criterion for the stability of motion of nonlinear system

As to an autonomous nonlinear system, the stability of the equilibrium state in a sufficiently small neighborhood of the equilibrium state can be determined by eigenvalues of the linear part of the nonlinear system provided that the eigenvalues are not in a critical case. Many methods may be used to detect the stability for a linear system. A lot of researches for determining the stability of a nonlinear system are completed by mathematicians and mechanicians but most of them are methods for the special forms of nonlinear systems. Till now, none of these methods can be conveniently applied to all nonlinear systems. The method introduced by this paper gives the necessary and sufficient conditions of the stability of a nonlinear system. The familiar Krasovski's method is a special case of this method.     

A result on a feedback system of ordinary differential equations

A monotone system of ordinary differential equations is considered. It is shown that the omega limit set of a bounded trajectory of this system contains an equilibrium point or a nonconstant periodic orbit. As an application, a four-dimensional system of ordinary differential equations of Lotka-Volterra type is presented. It is shown that if the interior equilibrium point of this system is unstable, then a periodic orbit is contained in the omega limit set of its bounded trajectories.     

A principle of objectivity for relativistic continuum mechanics

A principle of objectivity is formulated for Galilei and Minkowski space-time structure. In the Galilei case it is the principle of Noll. A reduced form for arbitrary local and causal functionals is obtained. For materials of grade n it coincides with the result of Bressan, and for simple materials with that of Bragg.     

On the relativistic thermodynamics of a continuous system in electromagnetic fields

Summary Whithout adopting a particular interaction between matter and fields, we deduce the relativistic equations of balance -for charged differential materials which move in electromagnetic fields. Successively, restriction for the relativistic constitutive equations are deduced by adopting a relativistic reduced dissipation inequality obtained from the 2nd principle of thermodynamics by P. M. Quan.

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Sommario Si determinano le equazioni relativistiche del bilancio per materiali differenziali carichi, con distribuzione continua di momento angolare interno, in moto in campi elettromagnetici senza ricorrere ad alcuna interazione particolare tra materia e campi. Successivamente si deducono delle restrizioni per le equazioni costitutive derivanti dall'imporre che esse rendano soddisfatta una diseguaglianza di dissipazione ridotta relativistica ottenuta dal2 ^o principio della termodinamica di P. M. Quan.

     

Maximum wave velocity in the moments system of a relativistic gas

We consider the system of moments associated with the relativistic Boltzmann-Chernikov equation. Using the particular symmetric form obtained by the closure procedure of Extended Thermodynamics we deduce a lower bound for the maximum velocity of wave propagation in terms of the number of moments for a non-degenerate gas. When the number of moments increases this velocity tends to the speed of light. We also give the lower bound estimate in the limit cases of ultrarelativistic fluids and in the non relativistic approximation. Received September 28, 1998     

On the criterion for the absolute stability of the control system

In this paper, firstly we give the criterion for the absolute stability of the second canonical form for the control system, including the equation of the longitudinal motions of a plane as a particular example. The corresponding result in [8], [9] is a particular example given in this paper. Secondly, we give the criteria for the absolute stability of the first canonical form in the usual case and in the critical case. Finally, we give some criteria for the absolute stability of the general form for the direct control system.All the results in this paper merely depend upon the relations between the parameters of the system itself to give an explicit algebraic discriminant.     

The new result on delayed finance system

In this paper, a finance system with time delay is considered. By linearizing the system at the unique equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the unique equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

分段线性耦合动力系统的周期解及稳定性分析

以某货车的主副钢板弹簧悬架为模型建立两自由度分段线性和轮胎非线性耦合动力学方程,采用打靶法求耦舍系统的周期解;将所得结果与近似解析法分析的结果进行比较,并利用Floquet理论判断周期解的稳定性。研究结果表明：当激励频率在等效固有频率附近时系统的周期解不稳定,其Floquet乘数的模大于1,系统振动剧烈,KBM法得到的周期解将产生较大误差;当路面工况突然发生改变或路面工况不变而载重发生较大变化时都有可能发生舅毡跃现象,造成系统的不稳定;周期解的长期时域图和Floquet理论验证了这一现象;     

Fluid models for relativistic electron beams

A system of equations is provided that may be used in the study of relativistic charged particle beams. The equations are based upon the equations of the kinetic theory for first, second and third order moments and the system is closed by letting the third order moment depend on the lower order ones. The form of that dependence is formally equal to the explicit constitutive function given by extended thermodynamics. However, here the contributions to the third order moment can be classed as being different in order of magnitude, because there is a smallness parameter characterizing the small dispersion of the particle beam. The resulting system of equations is quite specific, it is fully covariant and it is equivalent to a symmetric hyperbolic system thus ensuring existence and uniqueness of solutions.     

Necessary and sufficient criteria for absolute stability of the direct control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＲＩＴＥＲＩＡＦＯＲＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＴＨＥＤＩＲＥＣＴＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉ－ｙｅ（张继业）;ＳｈｕＺｈｏｎｇ－ｚｈｏｕ（舒仲周）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉ...     

A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids

In the context of heat conduction governed by the celebrated Cattaneo equation, Christov has recently proposed a modification of the time derivative term in order to satisfy the objectivity principle. For such a model applied to an incompressible fluid, the uniqueness of the solution is here proved.     

Optimal stability envelope of the damped hauger system

Summary By using the Frobenius' method, the authors obtain the virtually exact instability boundary of the externally damped viscoelastic Hauger system. The interaction between the two types of damping, yields results paradoxical to those obtained with each type of damping considered separately. An optimal stability envelope, which for a prescribed value of internal damping and any external damping gives the highest attainable critical load, is obtained. The addition of external damping for a prescribed internal damping, raises the critical load at best to the above optimal value; further addition of external damping diminishes the critical load. The Ziegler's jump or pseudo-destabilization due to infinitesimal internal damping, is shown to vanish with the introduction of external damping. For large external damping, the Ziegler's jump approaches a negative asymptotic value. These paradoxical results provide a deeper insight into the complex interaction between the internal and external damping in polygenic systems.

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Sommario Usando il metodo di Frobenius gli autori ottengono il contorno di instabilità virtualemnte esatto del sistema viscoelastico Hauger smorzato esternamente. La interazione fra i due tipi di smorzamento porta a risultati paradossali rispetto a quelli ottenuti con ogni tipo di smorzamento considerato separatamente. Si ottiene un inviluppo ottimale di stabilità che per dati valori degli smorzamenti interni ed esterni dà il massimo carico critico ottenibile. L'aggiunta di smorzamento esterno per un prescritto smorzamento interno aumenta il carico critico al massimo fino al sopra citato valore ottimale; un aumento ulteriore di smorzamento esterno diminuisce il carico critico. Si dimostra che con l'introduzione di smorzamento esterno diminuisce la discontinuità di Ziegler e la pseudo-instabilità dovuta a smorzamento interno infinitesimo. Per forti smorzamenti esterni la discontinuità di Ziegler tende a un valore asintotico negativo. Questi risultati paradossali forniscono una più approfondita comprensione della complessa interazione fra smorzamento interno ed esterno nei sistemi poligenici.

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