Stokes and Navier–Stokes equations with Navier boundary condition

Abstract

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω ⊂ R 3 of class C 1,1 . We prove the existence, uniqueness of weak and strong solutions in W 1,p (Ω) and W 2,p (Ω) for all 1 < p < ∞ considering minimal regularity on the friction coefficient α. Moreover, we deduce uniform estimates on the solution with respect to α which enables us to analyze the behavior of the solution when α → ∞.

On the Existence of Strong Solutions to a Fluid Structure Interaction Problem with Navier Boundary Conditions

“…In the case where η ∈ C 1,1 (ω) such a result is already known, see [1] (see also [4]). Here, we manage to obtain the result for η ∈ H 3 (ω) by following an idea of [14] and [15].…”

“…(6.46) We can then define the changes of variables X and Y by (3.3), and obtain similar estimates as in Lemma 6.3, Lemma 6.4 and Proposition 6.5. This yields Φ(f, g, h) Y∞ CR 2 , (6.47) and Φ(f (1) , g (1) , h (1) ) − Φ(f (2) , g (2) , h (2) ) Y∞ CR (f (1) , g (1) , h (1) ) − (f (2) , g (2) , h (2) ) Y∞ . (6.48) for (f, g, h), (f (i) , g (i) , h (i) ) ∈ B ∞,R .…”

“…(5.17) Let φ ∈ [H 1/2 (∂Ω)]3 such that φ n0 = 0, and let consider the system   −∇ • T( g, q) = 0 in Ω, ∇ • g = 0 in Ω, g = φ on ∂Ω.The above system admits a unique solution( g, q) ∈ [H 1 (Ω)] 3 × L 2 0(Ω) such that ∇ • g = 0 and g| ∂Ω = φ. This implies that (5.17) holds for all φ ∈ [H1 (Ω)] 3 , φ n0 = 0. Inserting (5.17) in (5.15) we get Ω 2νD(w) : D(φ)dy − Ω q∇ • φdy + β(w − T η 2 ) τ0 , φ τ0 H −1/2 ,H 1/2 = Ω (f − λw) • φdy,(5.18) for all φ ∈ [H 1 (Ω)] 3 , φ n0 = 0 .…”

“…Using the classical Korn inequality (see, for instance,[20]), the above relations imply that (u k ) k converges weakly to some u ∈ [H1 (Ω(η))]3 with D(u) = 0 and βu = 0 on ∂Ω(η). In particular, see [34, Lemma 1.1 p.18], there exist a, b ∈ R 3 , such that for any y ∈ Ω(η), u(y) = a + b ∧ y.…”

“…for all φ ∈ [H1 (Ω)]3 , ∇ • φ = 0, φ n0 = 0. Comparing(5.15) and(5.16) and taking into account thatΩ T(w, q) : D(φ)dy = 2ν Ω D(w) : D(φ)dy, ∀φ ∈ [H 1 (Ω)] 3 , ∇ • φ = 0, φ n0 = 0, we obtain − T(w, q)n 0 , φ H −1/2 ,H 1/2 = [β(w − T η 2 )] τ0 , φ H −1/2 ,H 1/2 = 0, ∀φ ∈ [H 1 (Ω)] 3 , ∇ • φ = 0, φ n0 = 0.…”