An extension of the principle of spatial averaging for inertial manifolds

Kwean, Hyuk-Jin

doi:10.1017/s1446788700036314

Cited by 17 publications

(19 citation statements)

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“…In particular, the principle of spatial averaging holds for an arbitrary rectangle Ω 2 ⊂ R 2 and for a cube Ω 3 ⊂ R 3 [11], although condition (2.4) is not guaranteed for the former and is violated for the latter. In [8], the existence of a…”

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confidence: 99%

On the hyperbolicity properties of inertial manifolds of reaction–diffusion equations

Romanov

2016

Dynamics of Partial Differential Equations

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Abstract. For 3D reaction-diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning absolutely normally hyperbolic inertial manifolds are discussed. Keywords: reaction-diffusion equations, inertial manifold, normal hyperbolicity. IntroductionThe existence of a smooth inertial manifold M for the dissipative parabolic equation in the infinite-dimensional Hilbert space implies [14,18,19] that its final dynamics (as t → +∞) is controlled by finitely many parameters. The additional property of normal hyperbolicity of the inertial manifold M guarantees the structural stability of this manifold. The stronger property of absolute normal hyperbolicity means one and the same hyperbolicity parameters for the entire M. So far, the existence of an inertial C 1 -manifold has been established for a rather narrow class of semilinear parabolic equations, while known examples of its nonexistence [2,15,16] seem to be somewhat artificial and are not related to problems of mathematical physics.The present paper deals with necessary conditions for the existence of the abovementioned two types of inertial manifolds of scalar and vector reaction-diffusion equations. For the 3D chemical kinetics equations with a cubic nonlinearity, we strive for 1 constructing examples separating the classes of problems admitting an inertial manifold, a normally hyperbolic inertial manifold, and an absolutely normally hyperbolic inertialmanifold. An example separating the first two possibilities is obtained for two-component systems. Namely, in Proposition 3.5 we construct an (uncoupled) system of such equations that has an inertial manifold but does not admit a normally hyperbolic inertial manifold. In particular, this system provides an example of an inertial manifold that is not normally hyperbolic. On the other hand, we present a system of two coupled reaction-diffusion equations of this type that do not admit a normally hyperbolic inertial manifold in the natural state space (Proposition 3.4). An example of a scalar 3D equation with a cubic nonlinearity without an absolutely normally hyperbolic inertial manifold is constructed. Note that the order of the polynomial nonlinearity in the chemical kinetics equations corresponds to the reaction order, which usually does not exceed 3. We also discuss how close the well-known sufficient conditions (the spectral jump condition and the spatial averaging principle) for the existence of strongly and weakly normally hyperbolic inertial manifolds are to being necessary.The paper is organized as follows. Section 1 contains elementary information about abstract semilinear parabolic equations. The necessary and suffi...

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confidence: 99%

On the hyperbolicity properties of inertial manifolds of reaction–diffusion equations

Romanov

2016

Dynamics of Partial Differential Equations

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“…With the property (2.29), the result follows from property (1) of Lemma 2.6 and the facts that the set of eigenfunctions of Laplace operator forms complete orthogonal basis for L 2 and that any bounded set of H 2 is a compact subset of L 2 for n ≤ 3. For more detail proof, we mention Kwean [1].…”

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confidence: 99%

“…Proof. Due to the invariant manifold theorem [2] and the dissipativity of (3.1), we can prove the existence of an inertial manifold ᏹ (see [1]). …”

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confidence: 99%

“…This problem was studied by Mallet-Paret and Sell [2] for 2-dimensional rectangular and 3-dimensional cubic domains and Kwean [1] extended their result into several new domains including the type of domains in (1.2). However for their works, they consider either Dirichlet or Neumann boundary conditions or periodic boundary conditions when the domain is a Cartesian product of intervals.…”

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confidence: 99%

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An inertial manifold and the principle of spatial averaging

Kwean

2001

International Journal of Mathematics and Mathematical Sciences

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“…Note that the proof of this theorem differs essentially from the one given in the first part of our study for the case of Dirichlet boundary conditions. In particular, we have to use a special cut-off procedure similar to the one developed in [10] (see also [5,9,20]) for the so-called spatial averaging method as well as the graph transform and invariant cones instead of the Perron method. The extra term u is added only in order to have dissipativity and the global attractor in the periodic case as well and is not essential for IMs.…”

Section: Introductionmentioning

confidence: 99%

Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions

Kostianko¹,

Zelik²

2018

Communications on Pure &Amp; Applied Analysis

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This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in [6] and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed by periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or nonexistence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.2000 Mathematics Subject Classification. 35B40, 35B45.

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An extension of the principle of spatial averaging for inertial manifolds

Cited by 17 publications

References 12 publications

On the hyperbolicity properties of inertial manifolds of reaction–diffusion equations

On the hyperbolicity properties of inertial manifolds of reaction–diffusion equations

An inertial manifold and the principle of spatial averaging

Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions

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