Existence Of Inertial Manifolds Research Articles

Inertial manifolds for functional differential equations with infinite delay

In this paper, we consider a class of functional differential equations with infinite delay in a real separable Hilbert space. By using the Lyapunov–Perron method, we prove the existence of inertial manifolds for mild solutions. The obtained results can be applied to Lotka–Volterra diffusion models with infinite delay.

Inertial Manifolds for Generalized Higher-Order Kirchhoff Type Equations

The existence of inertial manifolds for higher-order Kirchhoff type equations with strong damping terms is studied. The Hadamard graph norm conversion method is used for obtaining the existence of inertial manifolds for this kind of equations under certain spectral intervals.

Dynamics of a conserved phase-field system

Recently, in Bonfoh [Ann. Mat. Pura Appl. 2011;190:105–144], we investigated the dynamics of a nonconserved phase-field system whose singular limit is the viscous Cahn–Hilliard equation. More precisely, we proved the existence of the global attractor, exponential attractors, and inertial manifolds and we showed their continuity with respect to a singular perturbation parameter. In the present paper, we extend most of these results to a conserved phase-field system whose singular limit is the nonviscous Cahn–Hilliard equation. These equations describe phase transition processes. Here, we give a direct proof of the existence of inertial manifolds that differs from our previous method that was based on introducing a change of variables and an auxiliary problem.

Inertial Manifolds for 2D Generalized MHD System

In this paper, we prove the existence of inertial manifolds for 2D generalized MHD system under the spectral gap condition.

Inertial Manifolds for Weakly and Strongly Dissipative Hyperbolic Equations

One considers mixed boundary value problems for a quasilinear hyperbolic equation with a weak, as well as strong, dissipation. The nonlinear function in the equation is assumed Lipschitz continuous. For each of these problems one obtains the conditions on the Lipschitz constant that ensure the existence of inertial manifolds.

Long time dynamics of von Karman evolutions with thermal effects

 This paper presents a short survey of recent results pertaining to stability and long time behavior of von Karman thermoelastic plates. Questions such as uniform stability - and associated exponential decay rates for the energy function, existence of attractors in the case of internally/externally forced plates along with properties of attractors such as smoothness and dimensionality will be presented. The model considered consists of undamped oscillatory plate equation strongly coupled with heat equation. There are no other sources of dissipation. Nevertheless it will be shown that that the long-time behavior of the nonlinear evolution is ultimately finite dimensional and "smooth". In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter \gamma, which is known to change the character of dynamics from hyperbolic (\gamma > 0) to parabolic like (\gamma= 0). Other properties such as additional smoothness of attractors, upper-semicontinuity with respect to parameter \gamma and existence of inertial manifolds are also presented.

Long time dynamics of von Karman evolutions with thermal effects

 This paper presents a short survey of recent results pertaining to stability and long time behavior of von Karman thermoelastic plates. Questions such as uniform stability - and associated exponential decay rates for the energy function, existence of attractors in the case of internally/externally forced plates along with properties of attractors such as smoothness and dimensionality will be presented. The model considered consists of undamped oscillatory plate equation strongly coupled with heat equation. There are no other sources of dissipation. Nevertheless it will be shown that that the long-time behavior of the nonlinear evolution is ultimately finite dimensional and "smooth". In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter \gamma, which is known to change the character of dynamics from hyperbolic (\gamma > 0) to parabolic like (\gamma= 0). Other properties such as additional smoothness of attractors, upper-semicontinuity with respect to parameter \gamma and existence of inertial manifolds are also presented.

Spectral gaps, inertial manifolds and kinematic dynamos

Inertial manifolds are desirable objects when ones wishes a dynamical process to behave asymptotically as a finite-dimensional ones. Recently [Physica D 194 (2004) 297] these manifolds are constructed for the kinematic dynamo problem with time-periodic velocity. It turns out, however, that the conditions imposed on the fluid velocity to guarantee the existence of inertial manifolds are too demanding, in the sense that they imply that all the solutions tend exponentially to zero. The inertial manifolds are meaningful because they represent different decay rates, but the classical dynamos where the magnetic field is maintained or grows are not covered by this approach, at least until more refined estimates are found.

Spectral barriers and inertial manifolds for time- discretized dissipative equations

The existence of inertial manifolds for dissipative nonlinear evolutionary equationsunder time discretizations of the semi-implicit Euler method is proved by using a generalized concept of spectral barrier for time-discretized equations.

Inertial Manifolds and Forms for Stochastically Perturbed Retarded Semilinear Parabolic Equations

We construct inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. These inertial manifolds are finite-dimensional invariant surfaces, which attract exponentially all trajectories. We study the corresponding inertial forms, i.e., the restriction of the stochastic equation to the inertial manifold. These inertial forms are finite-dimensional Ito equations and they completely describe the long-time dynamics of the system under consideration. The existence of inertial manifolds and the properties of inertial forms allow us to show that under mild additional conditions the system has a global (random) attractor in the sense of the theory of random dynamical systems.

An inertial manifold and the principle of spatial averaging

We examine the existence of inertial manifold of a class of differential equations with particular boundary conditions.

Inertial manifolds with delay for retarded semilinear parabolic equations

We consider the system of parabolic equations with distributeddelay. The existence of Inertial Manifolds with Delay is proved. We prove that the system has finite number of determining modes and can be reproduced by a finite-dimensional system with concerntrated delays.

Differentiability of Parabolic Semi-Flows in Lp-Spaces and Inertial Manifolds

We consider the semi-flow defined by semi-linear parabolic equations in Lp-spaces and study the differentiable dependence on the initial data. Together with a spectral gap condition, this implies the existence of inertial manifolds in arbitrary space dimensions. The spectral gap condition is satisfied by the Laplace–Beltrami operator for a class of manifolds. The simplest examples are products of spheres.

Inertial manifolds for nonlinear viscoelasticity equations

In this paper, we investigate the dynamical behaviour of the following one-dimensional dispersive viscoelasticity equation: The nonlinear function is smooth, non-convex, unbounded and satisfies some general conditions of growth. By introducing equivalent norms, based on the Hamiltonian structure of the limit dispersive problem with , we can prove the existence of global absorbing balls, and a global attractor. Depending on the parameters , we make different transformations depending on whether is positive or negative; the latter case is the most interesting one as the dissipation is not strong enough to dominate dispersion. The semigroup S(t) generated by satisfies a spectral barrier property, and the existence of inertial manifolds is proved in both cases.

Global dynamics and control of a nonlinear body equation with strong structural damping

A nonlinear hinged extensible elastic body equation with strong structural damping and Balakrishnan-Taylor damping of full exponent is studied as a general model for large space structures of higher dimensions. In this paper, the absorbing sets and flat inertial manifold are obtained for this nonlinear body equation. The control spillover problem associated with the stabilization of this equation is resolved by constructing a linear finite dimensional feedback, control based on the existence of inertial manifolds of the uncontrolled equation. Moreover, the results obtained are robust with respect to the uncertainty in structural parameters.

A simple construction of inertial manifolds under time discretization

In this article, we obtain the existence of inertial manifolds under time discretization based on their invariant property. In [1], the authors gave their existence by finding the fixed point of some inertial mapping defined by a sum of infinite series: $ T_h^0\Phi(p)=\sum_{k=1}^{\infty}R(h)^kQF(p^{-k}+\Phi(p^{-k})) $ where $p^{-k}=(S^h_\Phi)^{-k}(p)$, see [1] for detailed definition. Here we get the existence by solving the following equation about $\Phi$: $\Phi(S_\Phi^h(p))=R(h)[\Phi(p)+hQF(p+\Phi(p))] \mbox{ for }\forall p\in PH.$ See section 1 for further explanation which describes just the invariant property of inertial manifolds. Finally we prove the $C^1$ smoothness of inertial manifolds.

Inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise

We present a construction of inertial manifolds and forms for semilinear parabolic equations subjected to additive white noise. The existence of inertial manifolds allow us to establish the existence of unique invariant distribution. We also derive the connection between this distribution and inertial forms of the system under consideration.

Inertial manifolds for Burgers' original model system of turbulence

The existence of inertial manifolds for Burgers' original mathematical model system of turbulence is investigated. The system consists of two equations and enjoys the characteristic quantity: the Reynolds number. Our object in this article is to express the existence in terms of this Reynolds number. The difficulty of first order derivatives is circumvented by the method originally due to M. Kwak.