Slow manifolds for infinite-dimensional evolution equations

Hummel, Felix; Kuehn, Christian

doi:10.4171/cmh/527

Cited by 4 publications

(17 citation statements)

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“…we observe that in the limit ε → 0 the expression εBv ε might not be well-defined as B is an unbounded operator. For more details and an example we refer to [29,Section 3.3]. Therefore, we introduce a splitting in the slow variable space • The spaces…”

Section: Assumptionsmentioning

confidence: 99%

“…As a first step we rewrite the system into the modified fast-slow system while maintaining the splitting in the slow variable v, where we replace the operator A by 1 ε Ãε and the nonlinear function f by f . Next, we apply the existence result shown in [29,Prop. 5.1], where we set…”

Section: Existence Of Slow Manifoldsmentioning

confidence: 99%

“…Different approaches such as energy and entropy methods [3,39], variational methods [13,37,40] or homogenization techniques [36,6] have been proposed in recent years. Our approach follows a recent geometric viewpoint using dynamical systems methods as introduced in [29] for fast-slow systems. Here, the idea is to generalize the theory developed by Fenichel [20,19] for ODE systems and combine it with techniques for infinite-dimensional Banach spaces cf.…”

Section: Introductionmentioning

confidence: 99%

“…[8,12,26] and the references therein. The challenges and also several applications of this approach can be found in [16,15,29]. Yet, the case of fast-reaction system is not covered by the results in [29] and here we cover this case.…”

Section: Introductionmentioning

confidence: 99%

“…The challenges and also several applications of this approach can be found in [16,15,29]. Yet, the case of fast-reaction system is not covered by the results in [29] and here we cover this case. More precisely, the fast-reaction evolution equations analyzed in this work are the following…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Assumptionsmentioning

confidence: 99%

Section: Existence Of Slow Manifoldsmentioning

confidence: 99%