Weak solutions for the exterior Stokes problem in weighted Sobolev spaces

Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem

“…is an isomorphism. Then by Lemma A.2(ii) the bounded bilinear form b(·, ·) : 3 , we conclude that the mixed variational formulation (4.3) has a unique solution (u, p) ∈ H 1 (R 3 ) 3 × L 2 (R 3 ) and there exists a constant C = C(µ) > 0 such that (u, p) satisfies inequality (4.4).…”

“…In addition, the operator in (4.9) is surjective (cf. [2, Proposition 2.1], [46, Proposition 2.4]), and hence the operator 3 , we conclude that the mixed variational formulation (4.3) has a unique solution (u, p) ∈ H 1 (R 3 ) 3 × L 2 (R 3 ) and there exists a constant C = C(µ) > 0 such that (u, p) satisfies inequality (4.4).…”

“…Similar to definition (2.2), H 1 (Ω ′ ) 3 and H 1 (Ω ′ ) 3 are the spaces of vectorvalued functions whose components belong to the scalar spaces H 1 (Ω ′ ) and H 1 (Ω ′ ), respectively (see, e.g., [38]). The Sobolev space H 1 (Ω ′ ) can be identified with the closureH 1 (Ω ′ ) of D(Ω ′ ) in the norm of H 1 (Ω ′ ) (see, e.g., [42, (3.11), (3.13)], [38,Theorem 3.33]).…”

Exterior Stokes problem in the half-space

“…Proof We easily show that there exist extensions f ∈ L p (R n ) of f and h ∈ W 1, p 0 (R n ) of h in R n and, thanks to Theorem 3.10 of [3], there exists (v, η) ∈ W 2, p…”

Localization of small obstacles in Stokes flow

“…First, we recall the definition of the weighted Sobolev spaces. We introduce the weight function ρ(x) := (2 + |x| 2 ) 1/2 and the following Sobolev spaces (for more details, see [3]): Notice that these spaces are reflexive Banach spaces with respect to the norms:…”