Operator splitting for dissipative delay equations

Bátkai, András; Csomós, Petra; Farkas, Bálint

doi:10.1002/pamm.201410475

Cited by 4 publications

(4 citation statements)

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“…In this way we can separate the delay part, and we only need to solve a delayed ordinary differential equation and a decoupled abstract Cauchy problem. The corresponding Lie-Trotter products are stable because the space can be renormed, if the delay operator contains the point evaluation at −1, such that both semigroups become contractive, see [9], and also [2].…”

Section: Delay Equationsmentioning

confidence: 99%

“…We study the stability of Lie-Trotter products of such matrix semigroups, and present three classes of examples (abstract delay equations, abstract inhomogeneous equations, abstract dynamic boundary value problems) and some open problems. This survey is based on the papers [1], [2] and [5], to which we refer the interested reader for more details and extensive bibliographical information. The Lie-Trotter product formula provides the motivation and the fundamental background for operator splitting schemes in numerical analysis.…”

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confidence: 99%

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Stability for Lie–Trotter products for some operator matrix semigroups

Bátkai

Csomós

Engel

et al. 2014

Proc Appl Math and Mech

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Coupled systems of linear differential equations in Banach spaces can be often handled by the theory of C0-semigroups of operator matrices. We study the stability of Lie-Trotter products of such matrix semigroups, and present three classes of examples (abstract delay equations, abstract inhomogeneous equations, abstract dynamic boundary value problems) and some open problems. This survey is based on the papers [1], [2] and [5], to which we refer the interested reader for more details and extensive bibliographical information.The Lie-Trotter product formula provides the motivation and the fundamental background for operator splitting schemes in numerical analysis. Given linear operators A, B on a Banach space E one is interested in the solution of the Cauchy probleṁ u(t) = (A + B)u(t) with fixed initial value u(0) = u 0 . The general theory of C 0 -semigroups yields that the problem is well-posed if and only if A + B is the generator of a C 0 -semigroup (U (t)) t≥0 , and that in this case the unique solution is given by u(t) = U (t)u 0 (we refer to standard monographs on semigroup theory, such as [7]). Now suppose that A, B generate the C 0 -semigroups (S(t)) t≥0 , (T (t)) t≥0 , respectively. If (λ − A − B) has dense range and domain D(A + B), then one has the following equivalence, stating the Lie-Trotter, or sequential splitting, formula: Stability:There are M, ω ≥ 0 with S t n T t n n ≤ M e ωt for all t ≥ 0, n ∈ N.The expressions S t n T t n n are called Lie-Trotter products and, as we see, their convergence is equivalent to their stability.This is an instance of the Lax equivalence theorem, and because of this equivalence we shall concentrate on stability issues. Every C 0 -semigroup (T (t)) t≥0 obeys an exponential estimate of the kind T (t) ≤ M e ωt for some M, ω ≥ 0, and is called quasicontractive if M can be take to be 1. If A, B both generate quasicontractive semigroups, then the stability of the corresponding Lie-Trotter products is a simple consequence of the submultiplicativity of the operator norm. In some cases the space E can be renormed such that both semigroups become quasicontractive, in which case the stability is again immediate. This is certainly so if B ∈ L (E), i.e., if B is bounded, since we can renorm E such that the semigroup (S(t)) t≥0 generated by A becomes quasicontractive, but a semigroup with bounded generator is quasicontractive.Proposition 1 (see [1, Prop. 2.4]) Let A be a generator and let B be bounded, then the corresponding Lie-Trotter products are stable.The importance of splitting procedures becomes inevitable if one considers Cauchy problems (on products E ×F of Banach spaces) given by operator matriceṡIn general, A can be represented only formally as an operator matrix: Its domain is usually nondiagonal (i.e., contains a certain coupling of the two coordinates). That is why, even if C = 0 and D = 0 the semigroup generated by A may not be of diagonal but rather of upper triangular form. Here is a characterization of such matrix semigroups:Proposition 2 (see [1, Prop. 2.1]) Le...

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Section: Delay Equationsmentioning

confidence: 99%

mentioning

confidence: 99%

Stability for Lie–Trotter products for some operator matrix semigroups

Bátkai

Csomós

Engel

et al. 2014

Proc Appl Math and Mech

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“…The literature both on the functional and the numerical analysis sides are extremely extensive, see, e.g., the surveys [32], [16], [17]. The decomposition of the compound problem can be based on various things, such as: on physical grounds (say, separating advection and diffusion phenomena, e.g., [24]), by mathematical-structural reasons (separating linear and non-linear parts, see e.g., [21,22], [19]; separating the history and present in case of delay equations, see [2]), etc. The starting point for exponential splitting methods is the definition of the mild solution of problem (1), that is the variation-of-constants formula: u(t) = e tA u(0) + t 0 e (t−τ )A g(τ, u(τ )) dτ.…”

Section: Introductionmentioning

confidence: 99%

On exponential splitting methods for semilinear abstract Cauchy problems

Farkas¹,

Jacob²,

Schmitz³

2022

Preprint

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Due to the seminal works of Hochbruck and Ostermann [21], [22] exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than π 2 (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing on the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of T. Kato [25] to the local solvability of the Navier-Stokes equation, where the L p -L r smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Ostermann and Hochbruck), or more generally in the setting of [21], but also allows the consideration of examples, such as non-analytic Ornstein-Uhlenbeck semigroups or the Navier-Stokes flow around rotating bodies.

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“…It was shown by Carrillo, Gwiazda and Ulikowska in [8] that properties of complicated models, like structured population models, can be obtained by splitting the original model into simpler ones and analyzing them separately, which also leads to switching schemes of Lie-Trotter form. Bátkai, Csomós and Farkas investigated Lie-Trotter product formulae for abstract nonlinear evolution equations with delay in [4], a general product formula for the solution of nonautonomous abstract delay equations in [5] and analyzed the convergence of operator splitting procedures in [3].…”

Section: Introductionmentioning

confidence: 99%

Lie-Trotter product formula for locally equicontinuous and tight Markov semigroup

Hille,

Ziemlanska

2018

Preprint

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In this paper we prove a Lie-Trotter product formula for Markov semigroups in spaces of measures. We relate our results to "classical" results for strongly continuous linear semigroups on Banach spaces or Lipschitz semigroups in metric spaces and show that our approach is an extension of existing results. As Markov semigroups on measures are usually neither strongly continuous nor bounded linear operators for the relevant norms, we prove the convergence of the Lie-Trotter product formula assuming that the semigroups are locally equicontinuous and tight. A crucial tool we use in the proof is a Schur-like property for spaces of measures. 2000 Mathematics Subject Classification. 37A30, 47D07, 47N40, 37M25. Key words and phrases. Lie-Trotter product formula, Markov semigroups, commutator conditions. The work of MZ has been partially supported by a Huygens Fellowship of Leiden University. * BL,d E(f ) * BL,d E(f ) ≤ max(1, |f | L,d M)tω(t) u Y

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Operator splitting for dissipative delay equations

Abstract: Operator splitting methods for a special class of nonlinear partial differential equations with delay are investigated.

Cited by 4 publications

References 26 publications

Stability for Lie–Trotter products for some operator matrix semigroups

Stability for Lie–Trotter products for some operator matrix semigroups

On exponential splitting methods for semilinear abstract Cauchy problems

Lie-Trotter product formula for locally equicontinuous and tight Markov semigroup

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