Tamtam Proceedings - lamsin

3D Navier-Stokes equations 52.1. The linear Stokes operatorLet us recall some results similar to those established in [1] for the associated Stokesproblem :νcurlω + ∇p = g, ω = curlu, divu = 0 in Ω, (4)satisfying the boundary conditions⎧⎨⎩u · n = 0 , u × n = 0 on Γ 1 ,p = µ , u × n = 0 on Γ 2 ,u · n = 0 , ω × n = λ on Γ 3 ,where we take g ∈ L 4 3 (Ω), λ ∈ L 2 (Γ 3 ) and µ ∈ L 2 (Γ 2 ).The mixed variational formulation associated with problem [4]-[5] is :⎧⎨⎩F ind (σ, u) ∈ X × M such thata(σ, τ) + b(τ, u) = 0 ∀τ ∈ X,b(σ, v) = −l(v) ∀v ∈ M,(5)(6)where, for all σ = (ω, p), τ = (θ, q) ∈ X and v ∈ M :a(σ, τ) = ν ∫ Ω ω.θdΩ , b(τ, v) = −ν ∫ Ω θ.curlvdΩ + ∫ Ω qdivvdΩ,l(v) = ∫ Ω g.vdΩ + ν ∫ Γ 3λ.v × ndΓ − ∫ Γ 2µv · ndΓ.The following Hilbert spaces are employed :M = {v ∈ H(div, curl; Ω); v · n| Γ1∪Γ 3= 0, v × n| Γ1∪Γ 2= 0},X = L 2 (Ω) × L 2 (Ω),where H(div, curl; Ω) = {v ∈ L 2 (Ω); divv ∈ L 2 (Ω), curlv ∈ L 2 (Ω)}.The space M is normed by ‖v‖ M= (‖v‖ 2 0,Ω + ‖divv‖2 0,Ω + ‖curlv‖2 0,Ω )1/2 .We introduce the seminorm |v| M= (‖divv‖ 2 0,Ω + ‖curlv‖2 0,Ω )1/2 and we assume that:• the seminorm |·| Mis equivalent to the norm ‖·‖ Min M,• M is compactly embedded in L p (Ω) with p > 4 ,• the traces of the elements of M belong to L 2 (Γ).For every (g, λ, µ) ∈ L 4 3 (Ω) × L 2 (Γ 3 ) × L 2 (Γ 2 ), problem (6) satisfies the Babuska-Brezzi conditions (cf. [2] for instance). Then we can define the linear and continuousStokes operator S :S : L 4 3 (Ω) × L 2 (Γ 3 ) × L 2 (Γ 2 ) → X × L 4 (Ω) with S(g, λ, µ) = (σ, u). (7)TAMTAM –Tunis– 2005