On the three‐dimensional stationary exterior Stokes problem with non standard boundary conditions

Abstract

We study the three‐dimensional stationary exterior Stokes problem with non standard boundary conditions corresponding to a slip‐without‐friction boundary conditions. Because the flow domain is unbounded, we set the problem in weighted spaces in order to control the behavior at infinity of solutions. This functional framework allows to prescribe various behaviors at infinity of the solutions. The established results are related to the existence and the uniqueness of strong and very weak solutions. Our strategy relies on the fact that due to the boundary conditions, the pressure and the velocity can be decoupled and, as a result, we solve two separate systems to find these quantities.

Very weak solution for the exterior stationary Stokes equations with Navier slip boundary condition

“…Proof. The proof of the following theorem is similar to [30,Theorem 7]. Observe that, in view of Proposition 3.5, if the pair (u, π) that belongs to W 0,2 −k−1 (Ω)×W −1,2 −k−1 (Ω) is a solution of (S T ), then for any (ϕ, q) ∈ S k+1 (Ω) × W 1,2 k+1 (Ω), adding (3.6) and (3.7), we have…”

“…The proofs of Lemma 2.3 and Lemma 2.4 can be found in [30]. Finally, in order to write a Green formula, we define for k ∈ Z:…”

“…We can just mention [34] where (1.2) was used for the stationary Navier-Stokes equations in exterior domains and [19] for the stationary exterior Stokes equations, the authors in their articles they studied some results of existence and uniqueness for different types of solutions, including the variational solution W 1,2 0 (Ω) × L 2 (Ω) and the strong solutions W 2,2 k+1 (Ω) × W 1,2 k+1 (Ω) for k ∈ Z . For the case of Navier boundary conditions without friction (α = 0), let us mention [8,30], where the following boundary conditions (also known as Hodge boundary conditions) were used:…”

“…The Stokes problem when Ω is bounded with Dirichlet boundary condition has been studied by a large number of authors, from different points of view, here so we give some examples: Giga and Sohr [23], Amrouche and Girault [5] Galdi, Simader and Sohr [20], K. Schumacher [35], Amrouche and Rodriguez-Bellido [14], for the exterior domain we can mention Amrouche and Meslameni [9]. The very weak solutions of the problem (1.1)-(1.2) has been studied for a bounded domain in [12] for α = 0, in [30] for the exterior domain for the case α = 0 and in [10] for the half-space.…”

$L^p$-theory for the exterior Stokes problem with Navier's type slip-without-friction boundary conditions

“…In the one hand, in the bounded domains, one can refer for instance to [26,8,13,15,16]. In the other hand, the case for the exterior domains, we can just mention [4,25]. The exterior problem (1.1)-(1.2), where as far as know, we can mention [33,29,19].…”