Stokes and Navier-Stokes equations with Navier boundary conditions

Tapia, P. Acevedo; Amrouche, Chérif; Conca, Carlos; Ghosh, Amrita

doi:10.1016/j.jde.2021.02.045

Cited by 30 publications

(25 citation statements)

References 29 publications

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“…The next theorem is an analogue of results, known on the Stokes problem with the homogeneous Dirichlet or Navier or Navier-type boundary conditions on the whole boundary of domain Ω, see [11], [2] and [3]. Recall that the theorem is non-trivial especially due to the variety of used boundary conditions and the fact that one cannot apply Riesz' theorem in the general L r -framework in order to establish the existence and uniqueness of a solution of the equation…”

Section: The Weak Stokes Operatormentioning

confidence: 86%

“…Nevertheless, also in the L r -setting, results on the existence and uniqueness of weak solutions of the Stokes problem with Dirichlet's boundary condition for the velocity can be found e.g. in [6], [29] and [11], with Navier's boundary condition in [2] and with the Navier-type boundary condition in [3]. Fundamental estimates have been basically obtained by means of hydrodynamical potentials in [6] (in 3D), [11], [2] and [3].…”

Section: Introductionmentioning

confidence: 99%

“…in [6], [29] and [11], with Navier's boundary condition in [2] and with the Navier-type boundary condition in [3]. Fundamental estimates have been basically obtained by means of hydrodynamical potentials in [6] (in 3D), [11], [2] and [3]. The 2D case (with Dirichlet's boundary condition and in a smooth domain) has also been solved in [29] by expressing the velocity by means of a stream function, application of operator ∇ ⊥ to the Stokes equation and the results on the biharmonic boundary-value problem.…”

Section: Introductionmentioning

confidence: 99%

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The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

Neustupa

2020

Preprint

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We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the L r -framework. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated by means of three types of boundary conditions: the conditions of periodicity on the "lower" and "upper" parts of the boundary, the Dirichlet boundary conditions on the "inflow" and on the profile and an artificial "do nothing"-type boundary condition on the "outflow". Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in W 1,r (Ω) and its continuous dependence on the data. We explain the sense in which the "do nothing" boundary condition on the "outflow" is satisfied.

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Section: The Weak Stokes Operatormentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

Neustupa

2020

Preprint

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“…(iv). Our idea of obtaining the global H 2 estimate is to decompose D into a series of bounded smooth domains Dk which only have three shapes, so that the related estimate constant in Dk could be uniform with k. In each Dk , we establish the H 2 estimate of the solution via the known conclusions for the linear Stokes system with the Navier-slip boundary condition in [1]. Then we achieve the global H 2 estimate by summarizing those estimates in Dk .…”

Section: Strategiesmentioning

confidence: 99%

On Leray's problem in an infinite-long pipe with the Navier-slip boundary condition

Li¹,

Pan²,

Yang³

2022

Preprint

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The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms.First, the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced, which will be served as the asymptotic profile of the solution to the generalized Leray's problem at spacial infinity. Second, a solenoidal vector function defined in the whole pipe, satisfying the Navier-slip boundary condition, having the designated flux and equalling to the Poiseuille flow at large distance, will be carefully constructed. This plays an important role in reformulating our problem. Third, the energy estimates depend on a combined L 2 estimate of the gradient and the stress tensor of the velocity.

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“…In general, the maximum regularity property of solutions of the steady Stokes problem is mostly known if domain Ω is sufficiently smooth, see e.g. [ [3], [5] for problems with the Navier-type boundary condition and [2], [9], [6] for problems with Navier's boundary condition on the whole boundary. Concerning nonsmooth domains, we can cite [17], [19] and [7], where the authors considered the Stokes problem in a 2D polygonal domain with the Dirichlet boundary conditions, and the aforementioned paper [32], where the maximum regularity property of the Stokes problem (1.1)-(1.8) has been proven in the L 2 -framework.…”

Section: Introductionmentioning

confidence: 99%

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework

Neustupa¹

2020

Preprint

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The paper deals with the Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. We use results from [32] (the maximum regularity property in the L 2 -framework) and [33] (the weak solvability in W 1,r ), and extend the findings on the maximum regularity property to the general L r -framework (for 1 < r < ∞). Using the reduction to one spatial period Ω, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves Γ 0 and Γ 1 , the Dirichlet boundary conditions on Γ in and Γ P and an artificial "do nothing"-type boundary condition on Γ out (see Fig. 1). We show that, although domain Ω is not smooth and different types of boundary conditions "meet" in the vertices of ∂Ω, the considered problem has a strong solution with the maximum regularity property for "smooth" data. We explain the sense in which the "do nothing" boundary condition is satisfied for both weak and strong solutions.

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Stokes and Navier-Stokes equations with Navier boundary conditions

Cited by 30 publications

References 29 publications

The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

On Leray's problem in an infinite-long pipe with the Navier-slip boundary condition

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework

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