The Neumann problem for the planar Stokes system

Medková, Dagmar

doi:10.1007/s11565-012-0154-8

Cited by 5 publications

(2 citation statements)

References 14 publications

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“…Fundamental estimates have been basically obtained using the Stokes fundamental solution and the corresponding Green tensor in [6] and [10], respectively, also the inf–sup condition in [1] and [3]. The 2D case in a smooth domain has also been studied in [34] (with Dirichlet's boundary condition, expressing the velocity using a stream function, applying the operator

\nabla ^{\perp }

to the Stokes equation and using the results on the biharmonic boundary‐value problem) and in [26] (the Neumann problem, applying the theory of hydrodynamical potentials).…”

Section: Introductionmentioning

confidence: 99%

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

Neustupa

2022

Mathematische Nachrichten

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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 Lr$L^r$‐setup. The mathematical model used here is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated using 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 W1,r(Ω)$\mathbf {W}^{1,r}(\Omega )$ 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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\nabla ^{\perp }

to the Stokes equation and using the results on the biharmonic boundary‐value problem) and in [26] (the Neumann problem, applying the theory of hydrodynamical potentials).…”

Section: Introductionmentioning

confidence: 99%

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

Neustupa

2022

Mathematische Nachrichten

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“…and (1. [2], [5] for problems with the Naviertype boundary condition, [3], [6] for problems with Navier's boundary condition, [31] for the 2D Stokes problem with the Neumann boundary condition (i.e. prescribing the normal part of the stress tensor on the boundary) and [32] for the 2D Stokes problem, prescribing the normal component of velocity and the pressure on the boundary.…”

Section: Introductionmentioning

confidence: 99%

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade

Neustupa¹

2020

Preprint

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We deal with a steady Stokes-type problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes-type problem is formulated by means of the Stokes equation, equation of continuity and three types of boundary conditions: the conditions of periodicity on the 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 explain on the level of weak solutions the sense in which the last condition is satisfied. We show that, although domain Ω is not smooth and different types of boundary conditions meet in the corners of Ω, the considered problem has a strong solution with the so called maximum regularity property.

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$$L^q$$ L q -Solution of the Robin Problem for the Stokes System with Coriolis Force

Medková

2018

J. Math. Fluid Mech.

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The Neumann problem for the planar Stokes system

Cited by 5 publications

References 14 publications

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade

$$L^q$$ L q -Solution of the Robin Problem for the Stokes System with Coriolis Force

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