An Optimality Condition for Approximate Inertial Manifolds

Sell, George R.

doi:10.1007/978-1-4612-4346-5_10

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…If the AKA-effect is present, it requires to study, instead of a diffusive approximation (1.7), hyperbolic approximations as in [40] (see also [5,6,22] and references therein), because the timescale of the AKA-effect is 'much faster' than that of the eddy viscosity. To track the symmetry condition we use the dynamical systems approach (see, e.g., [13,38,49,50]), that is we view the modulation equations (1.4) and (1.6) as an infinite-dimensional system of ordinary differential equations for the Fourier coefficients of solutions.…”

Section: The Methods Of Proofmentioning

confidence: 99%

Modulational stability of cellular flows

Novikov¹

2003

Nonlinearity

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We present here the homogenization of the equations for the initial modulational (large-scale) perturbations of stationary solutions of the two-dimensional Navier-Stokes equations with a time-independent periodic rapidly oscillating forcing. The stationary solutions are cellular flows and they are determined by the stream function φ = sin x 1 / sin x 2 / + δ cos x 1 / cos x 2 / , 0 δ 1. Two results are given here. For any Reynolds number we prove the homogenization of the linearized equations. For small Reynolds number we prove the homogenization for the fully nonlinear problem. These results show that the modulational stability of cellular flows is determined by the stability of the effective (homogenized) equations.

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Section: The Methods Of Proofmentioning

confidence: 99%

Modulational stability of cellular flows

Novikov¹

2003

Nonlinearity

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“…The nonlinear CH equation is a continuous model for the description of the dynamics of pattern formation in a phase transition, see [12]. It is known, see [8], that the CH model is dissipative, more precisely, there exists a constant M such that the ball…”

Section: T H E Ch Equationmentioning

confidence: 99%

Reduction of dimension of approximate intertial manifolds by symmetry

Rodriguez-Bernal

Wang

1999

Bull. Austral. Math. Soc.

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“…However, this regularity result may be applied to give very regular Approximate Inertial Manifolds. For instance, following the ideas developed by Chen [7] and Sell [45], we can modify the linear term by broadening its spectral gaps so that the stronger spectral gap condition is satisfied.…”

Section: Higher-order Regularitymentioning

confidence: 99%

Inertial manifolds and normal hyperbolicity

Rosa

Témam

1996

Acta Appl Math

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Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold. (1991): 35B40, 35K22, 35K55. Mathematics Subject Classifications

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An Optimality Condition for Approximate Inertial Manifolds

Cited by 7 publications

References 17 publications

Modulational stability of cellular flows

Modulational stability of cellular flows

Reduction of dimension of approximate intertial manifolds by symmetry

Inertial manifolds and normal hyperbolicity

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