Tamtam Proceedings - lamsin

252 Hernane-Boukarianda 3 =−11 + γ − b [b′′ + ∼ y(γ ′′ − b ′′ 2)] +(1 + γ − b) 2 (γ′ − b ′ )[b ′ + ∼ y(γ ′ − b ′ )].The gradient operator becomes :⎛∼∇ b,γ =The equation (7) will be written :⎝∂ + −b′ − ∼ y(γ ′ −b ′ ) ∂∂ ∼ x (1+γ−b) 2 ∂ ∼ y1 ∂1+γ−b ∂ ∼ y⎞⎠ .γ( ∼ x) = −F ∣ ∣ 2 ∣∣∣2 [ ∼ ∼ ∣∣∣2∇b,γψ ( ∼ 2 ∂ψ∼x, 1) +1 + γ − b ∂ ∼ y (∼ x, 1)],where γ and b are searched in the space :⎧⎫⎨Bc2,λ (R)=⎩ v ∈ C2,λ (R)/ ∑supe ∣ c|x| ∣D xv(x) ∣ ⎬k < ∞⎭and ψ ∼ in the space :⎧⎨Bc 2,λ (Q) =⎩ v ∈ C2,λ (Q)/ supwhere 0 < λ < 1 and c > 0.Now we are able to state the main result :x∈R0≤k≤2supk+l≤2( |x|, ∼ ∼ y)∈Q⎫∣ ∣ ∣∣e c ∼ ∣∣ ∣ ⎬x ∣∣D k ∼x D l ∼ yv∣ < ∞⎭ ,Theorem 1. There exist ∼ c > 0 , there exist K < 1 , there exist an open ball B of radiusr 0 centered at the origin of Bc 2,λ (R)×Bc2,λ (R) where c ∈]0, ∼ c[ , and 0 < λ < 1, thereexist a neighbourhood V b of zero in Bc 2,λ (R), there exist a mapping g : V b −→ Bc2,λ (R) ofclass C 1 , such that :∀ F ∈]0, K[, {∀(b, γ) ∈ B, T 1 (b, γ) = 0} ⇔ {b ∈ V b , γ = g(b).}.4. References[1] ABERGEL F. et BONA J.L, A Mathematical Theory For Viscous, Free-Surface Flows Over aPerturbed Plane, Report NO. AM 81, February (1991).[2] CHARLES J. AMICK, Proprites Of Steady Navier-Stokes Solutions For Certain UnboundedChannels And Pipes, Nonlinear Analysis, Theory, Methods and applications, Vol. 2 No 6, pp.689-720, (1978).[3] FREDERIC HELEIN, Ecoulement stationnaire dans un canal a fond presque plat,Groupe HydrodynamiqueNavale, URA CNRS 853, ENSTA, Centre de l’Yvette, 91120 Palaiseau, France.TAMTAM –Tunis– 2005