Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

Weis, Lutz

doi:10.1007/pl00004457

Cited by 632 publications

(681 citation statements)

References 8 publications

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“…Roughly speaking, our proof of Theorem 1 is divided into the following three steps. First of all, we show L p -L q maximal regularity of solutions to the model problems in the whole space, in the half-space and in the whole space with interface x n = 0 by using the operator valued Fourier multiplier theorem due to Weis [20] and Denk, Hieber and Prüss [8]. The key observation for this is to show the R-boundedness of the family of solution operators to such model problems.…”

Section: Resultsmentioning

confidence: 91%

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Maximal regularity and viscous incompressible flows with free interface

Shimizu

2008

Parabolic and Navier–Stokes Equations

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Abstract. We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time L p -L q maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.1. Introduction. We consider a time dependent problem for the Navier-Stokes equations with free interface in a bounded domain Ω ⊂ R n (n ≥ 2). For example, if a bottle with stopper half filled with water is put under zero gravity, then the air forms a bubble completely surrounded by the water. Let us consider such a model.Let Ω + t ⊂ Ω be occupied by the fluid of viscosity µ + > 0 which is given only at the initial time t = 0, while for t > 0 it is to be determined. Γ t denotes the boundary of Ω +

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Section: Resultsmentioning

confidence: 91%

“…One of our main tools to show L p -L q maximal regularity of (4) is R-boundedness and an operator valued Fourier multiplier theorem which have recently been developed by Weis [20] and Denk, Hieber and Prüss [8]. In the rest of this section, we discuss the analytic semigroup approach to the initial-boundary value problem:…”

mentioning

confidence: 99%

Maximal regularity and viscous incompressible flows with free interface

Shimizu

2008

Parabolic and Navier–Stokes Equations

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“…It is closely related to the notion of R-boundedness that has proved to be essential in e.g. questions of maximal regularity [22], and in fact the two notions coincide on many common spaces. For more background on γ-boundedness see [21].…”

Section: 3mentioning

confidence: 99%

Convergence of subdiagonal Padé approximations of C 0-semigroups

Egert

Rozendaal

2013

J. Evol. Equ.

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Abstract. Let (rn) n∈N be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for −A the generator of a uniformly bounded C 0 -semigroup T on a Banach space X, the sequence (rn(−tA)) n∈N converges strongly to T (t) on D(A α ) for α > 1 2 . Local uniform convergence in t and explicit convergence rates in n are established. For specific classes of semigroups, such as bounded analytic or exponentially γ-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.

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“…During the past few years a theory of L p -multipliers for operator valued functions has been developed by means of the notion of R-boundedness of sets of operators, see for example [1,2,3,4,5,6,7,8,10,12,13,17,18,19]. This theory has been applied to study maximal regularity of certain abstract evolution equations.…”

Section: Introductionmentioning

confidence: 99%

“…This theory has been applied to study maximal regularity of certain abstract evolution equations. For instance, L. Weis has shown in [18] that maximal L p -regularity of the abstract Cauchy problem u (t) = Au(t) + f (t) for a.e. t ≥ 0, u(0) = 0,…”

Section: Introductionmentioning

confidence: 99%

On R-boundedness of Unions of Sets of Operators

Gaans

2006

Partial Differential Equations and Functional Analysis

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It is shown that the union of a sequence T 1 , T 2 , . . . of R-bounded sets of operators from X into Y with R-bounds τ 1 , τ 2 , . . ., respectively, is Rbounded if X is a Banach space of cotype q, Y a Banach space of type p, and ∞ k=1 τ r k < ∞, where r = pq/(q−p) if q < ∞ and r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and p = q. The power r is sharp. Each Banach space that contains an isomorphic copy of c 0 admits operators T 1 , T 2 , . . . such that T k = 1/k, k ∈ N, and {T 1 , T 2 , . . .} is not R-bounded. Further it is shown that the set of positive linear contractions in an infinite dimensional L p is R-bounded only if p = 2.

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Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

Cited by 632 publications

References 8 publications

Maximal regularity and viscous incompressible flows with free interface

Maximal regularity and viscous incompressible flows with free interface

Convergence of subdiagonal Padé approximations of C 0-semigroups

On R-boundedness of Unions of Sets of Operators

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