The time-dependent Stokes problem with Navier slip boundary conditions on Lp -spaces

Baba, Hind Al; Amrouche, Chérif; Rejaiba, Ahmed

doi:10.1515/anly-2015-0034

Cited by 6 publications

(13 citation statements)

References 13 publications

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“…Throughout this work we will denote by

A_{p}

the Stokes operator with pressure boundary conditions on

L_{σ}^{p} false(normalΩ false), 1 < p < \infty

. As described in [, Section 2], due to boundary conditions the pressure can be decoupled from the Stokes problem using the following Dirichlet problem

Δ π = div f in Ω, π = 0 on Γ .

Thus the Stokes operator

A_{p}

is a linear closed densely defined operator

A_{p} : D true(A_{p} true) \subset L_{σ}^{p} false(normalΩ false) \mapsto L_{σ}^{p} false(normalΩ false)

, where

D true(A_{p} true) = true{bold-italicu \in {bold-italicW}^{2, p} (Ω); div bold-italicu = 0 in normalΩ, bold-italicu \times bold-italicn = bold0 on normalΓ true},

\forall u \in D true(A_{p} true), A_{p} u = - Δ u in Ω .

…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Proof Thanks to [, Theorems 3.5–3.7] we know that Problem has a unique solution

u \in W^{2, p} false(normalΩ false)

satisfying estimate

{∥ u ∥}_{L^{p} false(normalΩ false)} \leq \frac{C (Ω, p)}{| λ |} {∥ f ∥}_{L^{p} false(normalΩ false)} .

Set

z = curl u

in Ω and observe that z satisfies

({, \begin{matrix} λ z - Δ z = curl f, & div z = 0 & in & Ω, \\ z \cdot n = 0, & curl z \times n = bold-italic bold0 & on & Γ . \end{matrix})

Using the result of [, Theorems 4.10–4.11], we deduce that z belongs to

W^{2, p} false(normalΩ false)

and satisfies the estimate:

{∥ z ∥}_{L^{p} false(normalΩ false)}

…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Thanks to the operator

- B_{∣ Y_{p}}

generates a bounded analytic semigroup on

{bold-italic bold-scriptY}_{p}

for

1 < p < \infty

.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…More recently, the author together with C. Amrouche, N. Seloula proved the analyticity of the Stokes semigroup with the pressure boundary conditions in

L^{p}

‐spaces for all

1 < p < \infty

. The proof is based on a deep study of the following complex resolvent of the Stokes operator

λ u - Δ u + \nabla π = f, div u = 0 in Ω

with the boundary condition .…”

Section: Introductionmentioning

confidence: 99%

“…The proof is based on a deep study of the following complex resolvent of the Stokes operator

λ u - Δ u + \nabla π = f, div u = 0 in Ω

with the boundary condition . This results was exploited later in to treat the time dependent problem using a semigroup approach.…”

Section: Introductionmentioning

confidence: 99%

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Fractional powers of the Stokes operator with boundary conditions involving the pressure

Baba

2019

Mathematische Nachrichten

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Since the pioneer work of Leray [23] and Hopf [17], Stokes and Navier–Stokes problems have been often studied with Dirichlet boundary condition. Nevertheless, in the opinion of engineers and physicists such a condition is not always realistic in industrial and applied problems of origin. Thus arises naturally the need to carry out a mathematical analysis of these systems with different boundary conditions, which best represent the underlying fluid dynamic phenomenology. Based on the study of the complex and fractional powers of the Stokes operator with pressure boundary condition, we carry out a systematic treatment of the Stokes problem with the corresponding boundary conditions in Lp‐spaces.

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“…Throughout this work we will denote by

A_{p}

the Stokes operator with pressure boundary conditions on

L_{σ}^{p} false(normalΩ false), 1 < p < \infty

. As described in [, Section 2], due to boundary conditions the pressure can be decoupled from the Stokes problem using the following Dirichlet problem

Δ π = div f in Ω, π = 0 on Γ .

Thus the Stokes operator

A_{p}

is a linear closed densely defined operator

A_{p} : D true(A_{p} true) \subset L_{σ}^{p} false(normalΩ false) \mapsto L_{σ}^{p} false(normalΩ false)

, where

D true(A_{p} true) = true{bold-italicu \in {bold-italicW}^{2, p} (Ω); div bold-italicu = 0 in normalΩ, bold-italicu \times bold-italicn = bold0 on normalΓ true},

\forall u \in D true(A_{p} true), A_{p} u = - Δ u in Ω .

…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Proof Thanks to [, Theorems 3.5–3.7] we know that Problem has a unique solution

u \in W^{2, p} false(normalΩ false)

satisfying estimate

{∥ u ∥}_{L^{p} false(normalΩ false)} \leq \frac{C (Ω, p)}{| λ |} {∥ f ∥}_{L^{p} false(normalΩ false)} .

Set

z = curl u

in Ω and observe that z satisfies

({, \begin{matrix} λ z - Δ z = curl f, & div z = 0 & in & Ω, \\ z \cdot n = 0, & curl z \times n = bold-italic bold0 & on & Γ . \end{matrix})

Using the result of [, Theorems 4.10–4.11], we deduce that z belongs to

W^{2, p} false(normalΩ false)

and satisfies the estimate:

{∥ z ∥}_{L^{p} false(normalΩ false)}

…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Thanks to the operator

- B_{∣ Y_{p}}

generates a bounded analytic semigroup on

{bold-italic bold-scriptY}_{p}

for

1 < p < \infty

.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…More recently, the author together with C. Amrouche, N. Seloula proved the analyticity of the Stokes semigroup with the pressure boundary conditions in

L^{p}

‐spaces for all

1 < p < \infty

. The proof is based on a deep study of the following complex resolvent of the Stokes operator

λ u - Δ u + \nabla π = f, div u = 0 in Ω

with the boundary condition .…”

Section: Introductionmentioning

confidence: 99%

“…The proof is based on a deep study of the following complex resolvent of the Stokes operator

λ u - Δ u + \nabla π = f, div u = 0 in Ω

with the boundary condition . This results was exploited later in to treat the time dependent problem using a semigroup approach.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Baba

2019

Mathematische Nachrichten

Self Cite

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Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Guo

Kučera

Skalák

2022

Math Methods in App Sciences

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We deal with the conditional regularity of the weak solutions to the Navier–Stokes equations. We discuss a famous criterion by Vasseur in terms of divfalse(bold-italicufalse/false|bold-italicufalse|false) and extend this criterion to bounded domains with Navier and Navier‐type boundary conditions. Inspired by the equality bold-italicu·∇false|bold-italicufalse|λ=−λfalse|bold-italicufalse|λ+1divfalse(bold-italicufalse/false|bold-italicufalse|false),0.1emλ≥1, we further prove an optimal regularity criterion in terms of bold-italicu·∇false|bold-italicufalse|λ both for the whole three‐dimensional space and bounded domains with Navier's, Navier‐type, and Dirichlet boundary conditions. It specially means for λ=2 that the control of the energy flow in the critical norms LtpLxr provides the regularity of solutions. This criterion is proved by two different techniques.

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Very weak solution for the stationary exterior Stokes equations with non‐standard boundary conditions in L^p‐theory

Dhifaoui

2022

Math Methods in App Sciences

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The time-dependent Stokes problem with Navier slip boundary conditions on Lp -spaces

Abstract: This paper deals with the time-dependent Stokes problem with Navier boundary conditions on

Cited by 6 publications

References 13 publications

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Very weak solution for the stationary exterior Stokes equations with non‐standard boundary conditions in L^p‐theory

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The time-dependent Stokes problem with Navier slip boundary conditions on Lp -spaces

Abstract: This paper deals with the time-dependent Stokes problem with Navier boundary conditions on

Cited by 6 publications

References 13 publications

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Very weak solution for the stationary exterior Stokes equations with non‐standard boundary conditions in Lp‐theory

Contact Info

Product

Resources

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Very weak solution for the stationary exterior Stokes equations with non‐standard boundary conditions in L^p‐theory