Inertial Manifolds Research Articles

Inertial Manifolds via Spatial Averaging Revisited

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 23 October 2020Accepted: 20 September 2021Published online: 04 January 2022Keywordsinertial manifolds, spatial averaging, reaction-diffusion equation, Cahn--Hilliard equation, hyperviscous Navier--Stokes equation, Leray-$\alpha$ modelAMS Subject Headings35B33, 35B40, 35B42, 35Q30, 76F20Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

The Family of Exponential Attractors and Inertial Manifolds for a Generalized Nonlinear Kirchhoff Equations

In this paper, we study the long-time behavior of a class of generalized nonlinear Kichhoff equation under the condition of n dimension. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup related to the initial-boundary value problem are proved, and then the existence of its exponential attractor is obtained. By extending the space E0 to Ek, a family of the exponential attractors of the initial-boundary value problem is obtained. In the second part, we consider the long-time behavior for a system of generalized Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectrum interval condition.

A Family of Inertial Manifolds for a Class of Generalized Kirchhoff-Beam Equations

In this paper, we deal with a class of generalized Kirchhoff-Beam equations. At first, we take advantage of Hadamard’s graph to get the equivalent form of the original equations. Then, the inertial manifolds are proved by using spectral gap condition. We gain main result is that the family of inertial manifolds are established under the proper assumptions of nonlinear terms M(s) and N(s).

A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations

In this paper, the global dynamics of a class of higher order nonlinear Kirchhoff equations under n-dimensional conditions is studied. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup associated with the initial boundary value problem are proved, and the existence of a family of exponential attractors is obtained. Then, by constructing the corresponding graph norm, the condition of a spectral interval is established when N is sufficiently large. Finally, the existence of the family of inertial manifolds is obtained.

A Family of Inertial Manifolds for a Class of Asymmetrically Coupled Generalized Higher-Order Kirchhoff Equations

In this paper, we study the inertial manifolds for a class of asymmetrically coupled generalized Higher-order Kirchhoff equations. Under appropriate assumptions, we firstly exist Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, then we prove the existence of a family of inertial manifolds by showing that the spectral gap condition is true.

Inertial manifolds for parabolic differential equations: The fully nonautonomous case

<p style='text-indent:20px;'>We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t> s. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators <inline-formula><tex-math id="M1">\begin{document}$ (A(t))_{t\in { \mathbb {R}}} $\end{document}</tex-math></inline-formula> generates an evolution family <inline-formula><tex-math id="M2">\begin{document}$ (U(t,s))_{t\ge s} $\end{document}</tex-math></inline-formula> satisfying certain dichotomy estimates, and the nonlinear forcing term <inline-formula><tex-math id="M3">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> satisfies the Lipschitz condition such that certain dichotomy gap condition holds.</p>

An Approximate Inertial Manifold (Aim) Based Closure for Turbulent Flows

An Approximate Inertial Manifold (Aim) Based Closure for Turbulent Flows

A Family of Inertial Manifolds of Coupled Kirchhoff Equations

In this paper, we study the long-time behavior of the solution of the initial boundary value problem of the coupled Kirchhoff equations. Based on the relevant assumptions, the equivalent norm on Ek is obtained by using the Hadamard graph transformation method, and the Lipschitz constant lF of F is further estimated. Finally, a family of inertial manifolds satisfying the spectral interval condition is obtained.

A Family of the Inertial Manifolds for a Class of Generalized Kirchhoff-Type Coupled Equations

The paper considers the long-time behavior for a class of generalized high-order Kirchhoff-type coupled equations, under the corresponding hypothetical conditions, according to the Hadamard graph transformation method, obtain the equivalent norm in space , and we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectral interval condition.

Approximate Inertial Manifold-Based Model Reduction and Vibration Suppression for Rigid-Flexible Mechanical Arms

The dynamic characteristics of the mechanical arm with a rigid-flexible structure are very complex. The reason is that it is a complex DPS (distributed parameter system) with infinite dimension and nonlinearity in essence due to the rigid-flexible coupling. So, accurately positioning and controlling the rigid-flexible mechanical arms could be difficult. Therefore, a model reduction method of rigid-flexible mechanical arms based on the approximate inertial manifold is put forward. To repress the residual vibration of the end of the mechanical arm, a feedforward control strategy is designed. The high-dimensional solution of the vibration equation of the rigid-flexible mechanical arms is projected into the complete space composed of orthogonal decomposition modes. By using Galerkin’s method, the system is simplified and the approximate solution is obtained through the interaction between high-order and low-order modes. The truncated finite mode is also used to construct a lowest-order dynamic model on the basis of approximate inertia manifold. Given the reduced-order rigid-flexible mechanical arms dynamic model, dynamic response analysis is conducted to optimize the target position error and end residual vibration. A limited number of sinusoidal signals approximately combine the input signal, by using the particle swarm optimization algorithm to optimize the input signal, and the amplitude of the sinusoidal signal is corrected. The simulation results depict the superiority of the proposed method, which greatly suppresses the end residual vibration of the mechanical arm and realizes the accurate positioning of the end of the mechanical arm. In addition, the hardware experimental device of the rigid-flexible mechanical arms is constructed, and the experimental verification of the above method is put into effect. The simulation results of angular displacement and end vibration of the reduced model are accordant which is shown by the experimental results of the hardware platform.

Inertial manifolds for a singularly non-autonomous semi-linear parabolic equations

This paper devotes to the existence of an N N -dimensional inertial manifold for a class of singularly, i.e. A ( t ) A(t) may degenerate to 0 0 at some time t t , non-autonomous parabolic equations ∂ t u + A ( t ) u = F ( t , u ) + g ( x , t ) , t > τ ; u | t = τ = u τ ( x ) , x ∈ Ω , \begin{equation*} \partial _{t}u+A(t)u=F(t,u)+g(x,t),\;t>\tau ;\; \; u|_{t=\tau }=u_{\tau }(x),\;x\in \Omega , \end{equation*} where A ( t ) ≥ 0 A(t)\geq 0 for any t ≥ τ t\geq \tau , and Ω ⊂ R d \Omega \subset \mathbb {R}^{d} is a bounded domain with smooth boundary. Since the operator A ( t ) A(t) may degenerate, a compatibility condition for the operator A ( t ) A(t) and the nonlinear term F ( t , u ) F(t,u) was proposed to construct the inertial manifolds.

Stationary approximations of inertial manifolds for stochastic retarded semilinear parabolic equations

A class of retarded semilinear parabolic equation driven by an additive Q-Wiener process is studied. We construct a process via the Wiener shift to provide stationary approximations of the noise, and show that solutions of stationary approximations converge pointwisely to solutions of the stochastic retarded semilinear parabolic equation. We also prove the existence of Ck,γ smooth inertial manifolds for the stationary approximations, and show that such inertial manifolds converge to the those of the stochastically perturbed retarded semilinear parabolic equation.

Inertial effects and long-term transport properties of particle motion in washboard potential

The motion of non-interacting Brownian particles in the washboard potential is investigated in inertial regimes, when the overdamped approximation does not yield accurate predictions. The analysis is based on homogenization methods, deriving closed-form expressions for the long-term transport properties, i.e. effective velocity and diffusion coefficient. Different reduced models of increasing complexity, improving the overdamped approximation, are developed starting from the basic assumption that the velocity variable can be split into an “almost” deterministic and a fully stochastic contribution. The almost deterministic velocity term can be estimated from a fully deterministic or from a stochastic slow inertial manifold. The latter approach provides accurate predictions for all the asymptotic transport properties.

Admissible inertial manifolds for neutral equations and applications

We study the existence of admissible inertial manifolds for parabolic neutral functional differential equations of the form where the linear differential operator A is positive definite and self-adjoint with a discrete spectrum, the difference operator F is a bounded linear operator, and the delay nonlinear operator f is φ-Lipschitz for φ belonging to an admissible function space defined on . Our method is based on Lyapunov–Perron's equations, duality estimates in admissible spaces and F-induced trajectories. An application to heat transfer with delays in materials with memory is also given to illustrate our results.

Learning a reduced basis of dynamical systems using an autoencoder.

Machine learning models have emerged as powerful tools in physics and engineering. In this work, we use an autoencoder with latent space penalization to discover approximate finite-dimensional manifolds of two canonical partial differential equations. We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV), and damped KdV equations. We show that the resulting optimal latent space of the K-S equation is consistent with the dimension of the inertial manifold. We then uncover a nonlinear basis representing the manifold of the latent space for the K-S equation. The results for the KdV equation show that it is more difficult to recover a reduced latent space, which is consistent with the truly infinite-dimensional dynamics of the KdV equation. In the case of the damped KdV equation, we find that the number of active dimensions decreases with increasing damping coefficient.

Remarks on the existence of an inertial manifold

Remarks on the existence of an inertial manifold

Frequency theorem for parabolic equations and its relation to inertial manifolds theory

We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G.R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R.A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field.

Publisher's Note: “Using approximate inertial manifold approach to model turbulent non-premixed combustion” [Phys. Fluids 33, 035125 (2021)

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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.

Smooth Inertial Manifolds for Neutral Differential Equations with Small Delays

In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay.