Analyticity of the semi-group generated by the Stokes operator with Navier-type boundary conditions on 𝐿^{𝑝}-spaces

Baba, Hind Al; Amrouche, Chérif; Escobedo, Miguel

doi:10.1090/conm/666/13337

Cited by 11 publications

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“…In two forthcoming papers we study the two Problems (4.12) and (4.13). Proceeding in a similar way as in [5] we prove that these two Problems have a unique solution u ∈ W 1,p (Ω) (respectively (u, π) ∈ W 1,p (Ω) × W 1,p (Ω)/R) that satisfy the estimate…”

Section: P -Theorymentioning

confidence: 63%

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Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p -Spaces

Baba¹,

Amrouche²,

Escobedo³

2016

Arch Rational Mech Anal

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In this article we consider the Stokes problem with Navier-type boundary conditions on a domain Ω, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypothesis on the initial data u 0 and external force f . Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers we deduce maximal L p − L q regularity for the inhomogeneous Stokes problem.

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Section: P -Theorymentioning

confidence: 63%

“…where C 2 (Ω, p) = M 2 κ 1 (Ω, p) for some constant M 2 > 0, which gives us estimate (7.6). Next, to prove estimate (7.7) we proceed in the same way as in the proof of the estimate (4.10) (see [5,Theorem 4.11] for the proof).…”

Section: (72)mentioning

confidence: 99%

Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p -Spaces

Baba¹,

Amrouche²,

Escobedo³

2016

Arch Rational Mech Anal

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“…We recall also the following lemma that plays an important role in the proof of the resolvent estimates in L p theory (see [3,Lemma 2.4] for the proof): LEMMA 2.1. Suppose that is of class C 1,1 and let u ∈ W 1, p ( ) such that u ∈ L p ( ).…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

“…Thanks to [3] we know that the Stokes operator associated with Problem (4.7) generates a bounded analytic semigroup on the space of divergence free functions whose normal components vanishes on and that…”

Section: ) Set Z = Curl U(t) We Can Verify That Z(t) Is a Solutionmentioning

confidence: 99%

Instationary Stokes problem with pressure boundary condition in $${L^{p}}$$ L p -spaces

Baba¹,

Amrouche²,

Seloula³

2016

J. Evol. Equ.

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“…By construction we have that Ψ 1 (ξ) = 0 only if 3 4 ≤ |ξ| (b, a) ≤ 4. Since |2 (b, a) x| (b, a) = 2|x| (b, a) the supremum above reduces to x ∈ R n+1 such that 1 4 ≤ 3 8 ≤ |x| (b, a) ≤ 2 ≤ 4.…”

Section: Time-periodic Anisotropic Function Spacesmentioning

confidence: 99%

Time-Periodic Solutions to the Equations of Magnetohydrodynamics with Background Magnetic Field

Möller¹

2020

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Contents p 2 (1 − 3 q) ∈ (0, 1). Note that their result does not include the solutions constructed by Akiyama. The focus of this thesis lies on the existence of time-periodic solution to (MHDT), i.e., we want to find solutions (u, H, p) such that u(t+T , x) = u(t, x), H(t+T , x) = H(t, x) and p(t+T , x) = p(t, x) n 2 T 0 R n f (t, x) e −ix•ξ−i 2π T kt dxdt. The resulting function F Gn [f ] is defined on Z × R n , and the purely periodic part satisfies F Gn [P ⊥ f ](0, ξ) = 0. To construct time-periodic functions, a combination of classical Fourier multiplier results and a transference principle can be applied to yield existence of solutions on T×R n. Afterwards, classical methods of reflection and localisation can be used to construct solutions in sufficiently smooth domains. Applications of this technique can for example be found in Celik and Kyed [20,21], Eiter and Kyed [30] or Kyed and Sauer [61]. The advantage of this approach is clear: One directly constructs time-periodic solutions and therefore avoids considerations of initial value problems or the concept of R-boundedness. But since v 2,(2,1) (q,p),2 (T × R n). As a next step we come back to the trace problem and we see the advantage of working with Triebel-Lizorkin spaces in this context. By the Paley-Wiener-Schwartz theorem it is well-known that the vi ∞ (Ω) ≤ κ. This type of behaviour of the constant is to be expected, see for example Galdi and Kyed [38, Lemma 2.4]. Using Banach's fixed-point theorem together with the stated estimate, we show existence of time-periodic solutions to (MHDE) without general smallness assumptions on B 1. Note that this includes all constant magnetic fields H 0 , which is analogue to the results for the Oseen equations from [38].

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Analyticity of the semi-group generated by the Stokes operator with Navier-type boundary conditions on 𝐿^{𝑝}-spaces

Cited by 11 publications

References 0 publications

Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p -Spaces

Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p -Spaces

Instationary Stokes problem with pressure boundary condition in $${L^{p}}$$ L p -spaces

Time-Periodic Solutions to the Equations of Magnetohydrodynamics with Background Magnetic Field

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