Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

DOI10.1007/BF01047831zbMath0692.35053MaRDI QIDQ583486

Edriss S. Titi, Ciprian Foias, George R. Sell

Publication date: 1989

Published in: Journal of Dynamics and Differential Equations (Search for Journal in Brave)

zbMATH Keywords

inertial manifoldnonlinear dissipative partial differential equations

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Nonlinear parabolic equations (35K55) Initial value problems for linear higher-order PDEs (35G10) Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65M99) Higher-order parabolic equations (35K25) Numerical methods for partial differential equations, boundary value problems (65N99) Dynamical systems and ergodic theory (37-XX)

Related Items

Tracking properties of trajectories on random attracting Sets, A holistic finite difference approach models linear dynamics consistently, On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations with Nondense Domain, INERTIAL MANIFOLD FAMILY OF HIGH-ORDER NONLINEAR KIRCHHOFF-TYPE EQUATION, Concurrent MultiParameter Learning Demonstrated on the Kuramoto--Sivashinsky Equation, Finite-dimensional behavior in dissipative partial differential equations, Asymptotic behaviour of solutions to nonlinear shells in a supersonic flow¹, Hyperdissipative Navier–Stokes Equations Driven by Time-Dependent Forces: Invariant Manifolds, On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation, Inertial manifolds for retarded semilinear parabolic equations, Formation of patterns in the Rayleigh-Bénard convection, Noisy dynamic simulations in the presence of symmetry: data alignment and model reduction, A finite difference scheme for computing inertial manifolds, A new model reduction method for nonlinear dynamical systems, On the effectiveness of the approximate inertial manifold -- a computational study, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, Some closure results for inertial manifolds, Low dimensional approximate inertial manifolds for the kuramoto-sivashinsky equation, Stability Analysis of the Nonlinear Galerkin Method, Regularity criteria for the Kuramoto-Sivashinsky equation in dimensions two and three, Counterexamples to regularity of Mañé projections in the theory of attractors, Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds, Inertial manifolds for the hyperviscous Navier-Stokes equations, Accelerating waveform relaxation methods with application to parallel semiconductor device simulation^∗, Parameterizing the global attractor of the navier-stokes equations by nodal values, Finite determining parameters feedback control for distributed nonlinear dissipative systems -- a computational study, One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations, Regularity of the inertial manifolds for evolution equations in admissible spaces and finite-dimensional feedback controllers, The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems, Approximate inertial manifolds for retarded semilinear parabolic equations., Strong and weak constraint variational assimilations for reduced order fluid flow modeling, Uniform-in-Time Error Estimates for the Postprocessing Galerkin Method Applied to a Data Assimilation Algorithm, On the Rate of Convergence of the Nonlinear Galerkin Methods, An applicatin of approximate inertial manifolds to a weakly damped nonlinear schrödinger equation, A model-reduction method for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty, Variational representation of the projection dynamics and random motion of highly dissipative systems, Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Nonlinear Galerkin methods: The finite elements case, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations, Nonlinear feedback control of parabolic partial differential equation systems with time-dependent spatial domains, Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, A concise proof of the geometric construction of inertial manifolds, Map invariance and the state reconstruction problem for nonlinear discrete-time systems, Arbitrarily accurate approximate inertial manifolds of fixed dimension., Preserving dissipation in approximate inertial forms for the Kuramoto- Sivashinsky equation, Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation, Feedback control of nonlinear dissipative systems by finite determining parameters -- a reaction-diffusion paradigm, Smooth invariant foliations in infinite dimensional spaces, A continuity property for the inverse of Mañé's projection, Inertial manifolds: The non-self-adjoint case, Inertial manifolds for the 3D modified-Leray-\(\alpha \) model with periodic boundary conditions, Approximate inertial manifolds for 2D Navier-Stokes equations, On approximate inertial manifolds for two and three dimensional turbulent flows, Finite-dimensional description of convective reaction-diffusion equations, Determining finite volume elements for the 2D Navier-Stokes equations, New results of general \(n\)-dimensional incompressible Navier-Stokes equations, Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equations, On the number of determining nodes for the 2D Navier-Stokes equations, Smoothness of inertial manifolds, STOCHASTIC INERTIAL MANIFOLDS FOR DAMPED WAVE EQUATIONS, Finite dimensional exponential attractors for semilinear wave equations with damping, Holistic discretization ensures fidelity to Burgers' equation, Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay., Model reduction on inertial manifolds for N-S equations approached by multilevel finite element method, Approximate inertial manifolds and effective viscosity in turbulent flows, Analytic study of shell models of turbulence, Discussion on: ``Attenuation of oscillations in Galerkin systems using center-manifold techniques, Exponential attractors and their relevance to fluid dynamics systems, Bifurcation computations on an approximate inertial manifold for the 2D Navier-Stokes equations, Do inertial manifolds apply to turbulence?, Persistence of invariant sets for dissipative evolution equations, Inertial manifolds and normal hyperbolicity, On approximate inertial manifolds to the Navier-Stokes equations, Integral manifolds of nonautonomous evolution differential equations, A determining form for the subcritical surface quasi-geostrophic equation, Robust output feedback control of quasi-linear parabolic PDE systems, Approximation theories for inertial manifolds, Induced trajectories and approximate inertial manifolds, \textit{A priori} analysis of reduced description of dynamical systems using approximate inertial manifolds, The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation, Variational approach to closure of nonlinear dynamical systems: autonomous case, A unified approach to compute foliations, inertial manifolds, and tracking solutions, Nonlinear control of incompressible fluid flow: Application to Burgers' equation and 2D channel flow, Model reduction on approximate inertial manifolds for NS equations through multilevel finite element method and hierarchical basis, Inertial Manifolds for Reaction Diffusion Equations in Higher Space Dimensions, A sharp condition for existence of an inertial manifold, Construction of inertial manifolds by elliptic regularization, Postprocessing Fourier spectral methods: The case of smooth solutions, Inertial Manifolds for Certain Subgrid-Scale $\alpha$-Models of Turbulence, A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case, Long-time behavior of a two-layer model of baroclinic quasi-geostrophic turbulence, An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations, A sufficient condition for the existence of approximate inertial manifolds containing the global attractor., Computational efficiency and approximate inertial manifolds for a Bénard convection system

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension
• Spectral barriers and inertial manifolds for dissipative partial differential equations
• Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors
• Inertial manifolds for nonlinear evolutionary equations
• Geometric theory of semilinear parabolic equations
• Global lyapunov exponents, kaplan-yorke formulas and the dimension of the attractors for 2D navier-stokes equations
• Inertial Manifolds for Reaction Diffusion Equations in Higher Space Dimensions
• Approximation theories for inertial manifolds
• Dissipative periodic processes