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Stokes equations in an exterior domain with Navier boundary conditions

Abstract : In this manuscript, we study the three-dimensional stationary Stokes equations set in a exterior domain. The problem describes the flow of a viscous and incompressible fluid past a bounded obstacle. The distinctif feature here relies on the fact that the obstacle is assumed to a rough boundary. As a result, the fluid may slip on the boundary of the obstacle and, to take into account this property, we use the Navier boundary conditions. On the one hand, They model the impermeability of the obstacle, and on the other hand, the fact that the tangential component of the fluid velocity on the obstacle is proportional to the stress tensor. This problem has been well studied when set in a bounded domain. The standard Sobolev spaces provides, in this case, an adequate functional framework for a complete study. Since in our case, the domain is unbounded, these spaces are not adapted since it is necessary to describe the behaviour of the solutions to infinity. Therefore, we choose to set the problem in weighted Sobolev spaces where the weights describe the behaviour at infinity of the function (growth or decay).In this work, we first start by performing the mathematical analysis in the Hilbert setting. The key point here is to establish variant weighted Korn’s inequalities in order to get the coercivity of the bilinear form associated to the variational formulation. Next, we proved the existence, uniqueness of strong and very weak solutions. Finally, we study the extension of some of thses results to a weightedL^p-theory.
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Submitted on : Monday, November 16, 2020 - 1:48:08 PM
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Anis Dhifaoui. Stokes equations in an exterior domain with Navier boundary conditions. Analysis of PDEs [math.AP]. Université Bourgogne Franche-Comté; Université de Sfax (Tunisie), 2020. English. ⟨NNT : 2020UBFCD009⟩. ⟨tel-03007518⟩

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