Analyticity of the semigroup generated by the Stokes operator inL r spaces

Giga, Yoshikazu

doi:10.1007/bf01214869

Cited by 327 publications

(231 citation statements)

References 17 publications

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“…We denote by Π the unbounded connected component of We denote by X p (Π) the closure of the space of divergence-free, C ∞ c (Π) vector fields with respect to the L p -norm. The Stokes operator in X p generates an analytic semigroup of class C 0 on X p (Π), for any 1 < p < ∞, see [13], so that, in particular, problem (2.1) is well-posed in X p (Π).…”

Section: Estimates For the Stokes Semigroupmentioning

confidence: 99%

Two-dimensional incompressible viscous flow around a small obstacle

2006

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Abstract. In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity ω 0 and the circulation γ of the initial flow around the obstacle.We prove that, if γ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity ω 0 + γδ, where δ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [15], where the effect of the small obstacle appears in the coefficients of the PDE and not only on the initial data. The main ingredients of the proof are L p − L q estimates for the Stokes operator in an exterior domain, a priori estimates inspired on Kato's fixed point method, energy estimates, renormalization and interpolation.

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Section: Estimates For the Stokes Semigroupmentioning

confidence: 99%

Two-dimensional incompressible viscous flow around a small obstacle

2006

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“…We may assume without loss of generality that 20) by using the A r -weightω j := ω j (Q) −1 ω j instead of ω j if necessary. Note that (3.20) also holds for r ′ , {ω ′ j } in the following form:…”

Section: Proof: Assume That This Lemma Is Wrong Then There Is a Consmentioning

confidence: 99%

“…There are many papers dealing with resolvent estimates ( [7], [8], [15], [16], [20]; see Introduction of [10] for more details) or maximal regularity (see e.g. [1], [14], [16]) of Stokes operators for domains with compact as well as noncompact boundaries.…”

Section: Introductionmentioning

confidence: 99%

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Farwig

2007

J. math. fluid mech.

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We study resolvent estimate and maximal regularity of the Stokes operator in L q -spaces with exponential weights in the axial directions of unbounded cylinders of R n , n ≥ 3. For straights cylinders we obtain these results in Lebesgue spaces with exponential weights in the axial direction and Muckenhoupt weights in the cross-section. Next, for general cylinders with several exits to infinity we prove that the Stokes operator in L q -spaces with exponential weight along the axial directions generates an exponentially decaying analytic semigroup and has maximal regularity.The proofs for straight cylinders use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the R-boundedness of the family of solution operators for a system in the cross-section of the cylinder parametrized by the phase variable of the onedimensional partial Fourier transform. For general cylinders we use cut-off techniques based on the result for straight cylinders and the result for the case without exponential weight.

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“…introduit par Giga [9] pour leséquations stationnaires de Stokes aété beaucoupétudié ces dernières années pour leséquations stationnaires de Navier-Stokes. Nous donnons ici une nouvelle preuve plus simple de l'existence de solution très faible pour leséquations stationnaires de Navier-Stokes, qui s'appuie sur des arguments de densité et un cadre fonctionnel approprié pour définir de manière plus rigoureuse les traces des champs de vecteurs peu réguliers.…”

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“…The notion of very weak solutions (u, q) ∈ L p (Ω)×W −1,p (Ω) for the stationary Stokes or Navier-Stokes equations, corresponding to very irregular data, has been developed in the last years by Giga [9] (and also by Lions-Magenes [11] for the Laplace's equation, in a domain Ω of class C ∞ ), Amrouche-Girault [1] (in a domain Ω of class C 1,1 ) and more recently by Galdi-Simader-Sohr [8], Farwig-Galdi-Sohr [7] (in a domain Ω of class C 2,1 , see also Schumacher [14]) and finally by Kim [10] (in a domain Ω of class C 2 with connected boundary). In this context, the boundary condition is chosen in L p (Γ) (see Brown-Shen [3], Conca [5], Fabes-Kenig-Verchota [6], Moussaoui [12], Shen [15], Savaré [13]) or more generally in W −1/p,p (Γ).…”

Section: Introductionmentioning

confidence: 99%

Very weak solutions for the stationary Stokes equations

Amrouche¹,

Rodríguez-Bellido²

2010

Comptes Rendus. Mathématique

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The concept of very weak solution introduced by Giga [9] for the stationary Stokes equations has been intensively studied in the last years for the stationary Navier-Stokes equations. We give here a new and simpler proof of the existence of very weak solution for the stationary Navier-Stokes equations, based on density arguments and an adequate functional framework in order to define more rigourously the traces of non regular vector fields. We also obtain regularity results in fractional Sobolev spaces. All these results are obtained in the case of a bounded open set, connected of class C 1,1 of R 3 and can be extended to the Laplace's equation and to other dimensions. To cite this article: C. Amrouche, M.A. Rodríguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005). RésuméSolutions très faibles pour leséquations stationnaires de Stokes. Le concept de solution très faible introduit par Giga [9] pour leséquations stationnaires de Stokes aété beaucoupétudié ces dernières années pour leséquations stationnaires de Navier-Stokes. Nous donnons ici une nouvelle preuve plus simple de l'existence de solution très faible pour leséquations stationnaires de Navier-Stokes, qui s'appuie sur des arguments de densité et un cadre fonctionnel approprié pour définir de manière plus rigoureuse les traces des champs de vecteurs peu réguliers. On obtient aussi résultats de régularité dans des espaces de Sobolev fractionnaires. Tous les résultats sont obtenus dans le cas d'un ouvert connexe de classe C 1,1 de R 3 et peuventêtreétendusà l'équation de Laplace ainsi qu'aux autres dimensions. Pour citer cet article : C. Amrouche, M.A. Rodríguez-Bellido, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

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Analyticity of the semigroup generated by the Stokes operator inL r spaces

Cited by 327 publications

References 17 publications

Two-dimensional incompressible viscous flow around a small obstacle

Two-dimensional incompressible viscous flow around a small obstacle

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Very weak solutions for the stationary Stokes equations

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