Navier-stokes equations in thin 3D domains with navier boundary conditions

We consider the Navier-Stokes equations on a thin domain of the form Ω_ε = {x ∈ ℝ³ | x₁, x ₂ ∈ (0, 1), 0 < x₃ < εg {x₁, x₂)} supplemented with the following mixed boundary conditions: periodic boundary conditions on the lateral boundary and Navier boundary conditions on the top and the bottom. Under the assumption that ∥u ₀∥_H1(Ωε ≤ C_ε ^1/2, par;Mu₀ⁱ∥_L2(Omega;ε) C for i ∈ {1,2} and similar assumptions on the forcing term, we show global existence of strong solutions; here u₀ⁱ denotes the i-th component of the initial data u₀ and M is the average in the vertical direction, that is, Mu₀ⁱ(x₁, x₂ = (εg)^-1 ∫₀^εg u₀ ⁱ(x₁, x₂, x₃) dx₃ Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as ε → 0, we prove convergence to a solution of a limiting system which is a Navier-Stokes-like equation where the function g plays an important role. Finally, we compare the attractor of the Navier-Stokes equations with the one of the limiting equation. Indiana University Mathematics Journal