An initial boundary-value problem for model electromagnetoelasticity system

In this paper we consider an initial boundary-value problem related to the electrodynamics of vibrating elastic media. The aim is to prove an existence and uniqueness result for a model describing the nonlinear interactions of the electromagnetic and elastic waves. We assume that the motion of the continuum occurs at velocities that are much smaller than the propagation velocity of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations, one of them is the hyperbolic equation (an analog of the Lamé system) and another one is the parabolic equation (an analog of the diffusion Maxwell system). One stability result is proved too.     

Stability to weakly dissipative Timoshenko systems

In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.     

General decay of solutions in one-dimensional porous-elastic system with memory

In this work we consider a one-dimensional porous-elastic system with memory effects. It is well-known that porous-elastic system with a single dissipation mechanism lacks exponential decay. In contrary, we prove that the unique dissipation given by the memory term is strong enough to exponentially stabilize the system, depending on the kernel of the memory term and the wave speeds of the system. In fact, we prove a general decay result, for which exponential and polynomial decay results are special cases. Our result is new and improves previous results in the literature.     

具有动态边界的非均匀Euler-Bernoulli梁的边界反馈镇定

本讨论一端带有重物的Euler-Bernoulli梁的边界反馈镇定问题。在生物的质量忽略不计而只考虑重物的转动惯量的情况下，证明了同时在梁的自由端施加力和力矩反馈，闭环系统的能量可被指数镇定。进而对于系统只有力反馈或只有力矩反馈的情况，得到了闭环系统（指数）稳定的充分必要条件。     

Stability and optimality of decay rate for a weakly dissipative Bresse system

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

The stability of travelling fronts for general scalar viscous balance law

This paper is concerned with the existence and stability of travelling front solutions for some general scalar viscous balance law. By shooting methods we prove the existence of some class of travelling fronts for any positive viscosity. Further by analytic semigroup theory and detailed spectral analysis, we show that the travelling fronts obtained are asymptotically stable in some appropriate exponentially weighted space. Especially for all sufficiently small viscosity, the travelling waves are proved to be uniformly exponentially stable in the same weighted space.     

多维双极Euler-Poisson方程的C~1解的整体存在性(英文)

本文研究了出现在半导体器件或者等离子中的多维双极Euler-Poisson方程,证明了它的初值问题的C1解在Besov空间的整体存在性,同时也得到了在二维和三维情形下,速度的璇度以指数的速率收敛到零.     

Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossing-monostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson??s blowflies equation in population dynamics and Mackey?CGlass model in physiology.     

Exact solitary water waves with capillary ripples at infinity

We prove the existence of solitary water waves of elevation, as exact solutions of the equations of steady inviscid flow, taking into account the effect of surface tension on the free surface. In contrast to the case without surface tension, a resonance occurs with periodic waves of the same speed. The wave form consists of a single crest on the elongated scale with a much smaller oscillation at infinity on the physical scale. We have not proved that the amplitude of the oscillation is actually nonzero; a formal calculation suggests that it is exponentially small.     

Lack of exponential stability to Timoshenko system with viscoelastic Kelvin–Voigt type

We study the Timoshenko systems with a viscoelastic dissipative mechanism of Kelvin–Voigt type. We prove that the model is analytical if and only if the viscoelastic damping is present in both the shear stress and the bending moment. Otherwise, the corresponding semigroup is not exponentially stable no matter the choice of the coefficients. This result is different to all others related to Timoshenko model with partial dissipation, which establish that the system is exponentially stable if and only if the wave speeds are equal. Finally, we show that the solution decays polynomially to zero as \({t^{-1/2}}\) , no matter where the viscoelastic mechanism is effective and that the rate is optimal whenever the initial data are taken on the domain of the infinitesimal operator.     

On a maximal inequality and its application to SDEs with singular drift

In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.     

Monotonicity,uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c ?c _?. Then we show that the traveling wave fronts with speed c >c _? are exponentially asymptotically stable.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

Multidimensional stability of planar waves for delayed reaction-diffusion equation with nonlocal diffusion

In this paper, we consider the multidimensional stability of planar waves for a class of nonlocal dispersal equation in $n$--dimensional space with time delay. We prove that all noncritical planar waves are exponentially stable in $L^{\infty}(\RR^n )$ in the form of $\ee^{-\mu_{\tau} t}$ for some constant $\mu_{\tau} =\mu(\tau)>0$( $\tau >0$ is the time delay) by using comparison principle and Fourier transform. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the critical planar waves, we prove that they are asymptotically stable by establishing some estimates in weighted $L^1(\RR^n)$ space and $H^k(\RR^n) (k \geq [\frac{n+1}{2}])$ space.     

The exponential behavior of the stochastic three-dimensional primitive equations with multiplicative noise

In this article, we study the stability of weak solutions to the stochastic three-dimensional (3D) primitive equations (PEs) with multiplicative noise. In particular, we prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.     

On the influence of gyroscopic forces on the stability of steady-state motion : PMM vol. 39, n 6, 1975, pp. 963–973

The question of the influence of gyroscopic forces on the stability of steady-state motion of a holonomic mechanical system when the forces depend upon the velocities of only the position coordinates was answered by the Kelvin-Chetaev theorems [1] on the influence of gyroscopic and dissipative forces on the stability of equilibrium. However, if the gyroscopic forces depend as well on the velocities of the ignorable coordinates, then their influence on the stability of steady-state motions can, as the two problems in [2] show, prove to be entirely different from the influence of gyroscopic forces depending only on the velocities of the position coordinates. In this paper we investigate the influence of gyroscopic forces depending linearly on the velocities of the generalized coordinates, including the ignorable ones, on the stability of the steady-state motion of a holonomic conservative system. We prove that when the gyroscopic forces applied with respect to the ignorable coordinates are given as total time derivatives of certain functions of the position coordinates, the gyroscopic forces can both stabilize as well as destabilize the steady-state motion. Under certain conditions, this influence is also preserved for the action of dissipative forces depending on the velocities of only the position coordinates. In the case of action of dissipative forces depending also on the velocities of the ignorable coordinates, we have indicated the stability and instability conditions of the steady-state motion. Examples are considered. In conclusion, we discuss the conditions under which the application of gyroscopic forces to the system is equivalent to adding terms depending linearly on the generalized velocities to the Lagrange function.     

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε ^?1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).     

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.     

Gauss point mass lumping schemes for Maxwell's equations

Finite element and finite difference methods for approximating the Maxwell system propagate numerical waves with slightly incorrect velocities, and this results in phase error in the computed solution. Indeed this error limits the type of problem that can be solved, because phase error accumulates during the computation and eventually destroys the solution. Here we propose a family of mass-lumped finite element schemes using edge elements. We emphasize in particular linear elements that are equivalent to the standard Yee FDTD scheme, and cubic elements that have superior phase accuracy. We prove theorems that allow us to perform a dispersion analysis of the two common families of edge elements on rectilinear grids. A result of this analysis is to provide some justification for the choice of the particular family we use. We also provide a limited selection of numerical results that show the efficiency of our scheme. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 63–88, 1998