Kwak Transform and Inertial Manifolds revisited

Kostianko, Anna; Zelik, Sergey

doi:10.1007/s10884-020-09913-9

Cited by 10 publications

(14 citation statements)

References 32 publications

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“…see [21], which coincides up to the square root in the right-hand side with the case of β = 1/2 in the self-adjoint case and very far from the expected condition with β = 0:…”

Section: Introductionmentioning

confidence: 66%

See 1 more Smart Citation

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

Chepyzhov¹,

Kostianko²,

Zelik³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.2000 Mathematics Subject Classification. 35B40, 35B45.

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“…see [21], which coincides up to the square root in the right-hand side with the case of β = 1/2 in the self-adjoint case and very far from the expected condition with β = 0:…”

Section: Introductionmentioning

confidence: 66%

“…This difference causes the crucial mistake in the attempt to construct the IM for the 2D Navier-Stokes equations using the so-called Kwak transform, see [21,22,37]. On the other hand, for the non-selfadjoint operator of the form…”

Section: Introductionmentioning

confidence: 99%

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

Chepyzhov¹,

Kostianko²,

Zelik³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…Recalling that the Weyl asymptotic for the eigenvalues of the Laplacian reads λ m ∼ Cm 2/d the above spectral gap condition (depending on the domain Ω ⊂ R d ) may only hold in 1 dimension. Methods to overcome this problem are discussed in [32].…”

Section: Assumptionsmentioning

confidence: 99%

Fast Reactions and Slow Manifolds

Kuehn¹,

Sulzbach²

2023

Preprint

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In this paper we generalize the Fenichel theory for attracting critical/slow manifolds to fast-reaction systems in infinite dimensions. In particular, we generalize the theory of invariant manifolds for fastslow partial differential equations in standard form to the case of fast reaction terms. We show that the solution of the fast-reaction system can be approximated by the corresponding slow flow of the limit system. Introducing an additional parameter that stems from a splitting in the slow variable space, we construct a family of slow manifolds and we prove that the slow manifolds are close to the critical manifold. Moreover, the semi-flow on the slow manifold converges to the semi-flow on the critical manifold. Finally, we apply these results to an example and show that the underlying assumptions can be verified in a straightforward way.• We show that the fast-reaction systems are well-posed and prove the convergence of solutions as the parameter ε → 0 (see Section 3).• We generalize Fenichel's theorem for fast-reaction systems in infinite-dimensional Banach spaces.Under suitable assumptions we prove the existence of a C 1 -regular slow manifold that is close to the critical manifold in a suitable sense. Moreover, we show that the critical manifold attracts solutions and that the flows on the slow manifold converge to the flow on the critical manifold. To conclude this part we apply the theory to an example (see Section 4).• Lastly we present a brief overview on the theory of C 0 semigroups and their application to abstract Cauchy problems (see Appendix A).

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“…Our general theorem is applicable not only for a scalar reaction-diffusion equation (6.1), but also for systems where the analogue of (6.2) is known, for instance, for the case of 1D complex Ginzburg-Landau equation (however, one should be careful in the case where the diffusion matrix is not self-adjoint and especially when it contains non-trivial Jordan cells. In this case, even Lipschitz IM may not exist, see [25] for more details).…”

Section: Examples and Concluding Remarksmentioning

confidence: 99%

Smooth extensions for inertial manifolds of semilinear parabolic equations

Kostianko,

Zelik

2021

Preprint

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The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than C 1,ε -regularity for such manifolds (for some positive, but small ε). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a C n -smooth inertial manifold (where n ∈ N is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the C 1,ε -smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem. Contents 1. Introduction 1 2. Preliminaries I: Taylor expansions and Whitney Extension Theorem 6 3. Preliminaries II: Spectral gaps and the construction of an inertial manifold 10 4. Main result 18 5. Verifying the compatibility conditions 25 6. Examples and concluding remarks 30 References 34

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Kwak Transform and Inertial Manifolds revisited

Cited by 10 publications

References 32 publications

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

Fast Reactions and Slow Manifolds

Smooth extensions for inertial manifolds of semilinear parabolic equations

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