Navier-Stokes equations in thin 3D domains with Navier boundary conditions Dragos IftimieGenevieve RaugelGeorge Sell 35Q3076D0546E3535B6535K55Navier-Stokes equationsthin domainglobal existenceglobal regularityglobal attractor We consider the Navier-Stokes equations on a thin domain of the form $\Omega_{\epsilon} = \{ x \in \mathbb{R}^3 \mid x_1,x_2 \in (0,1),\ 0 < x_3 < \epsilon g(x_1,x_2) \}$ 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 $\| u_0 \|_{H^1(\Omega_{\epsilon})} \leq C\epsilon^{-1/2}$, $\| Mu^i_0 \|_{L^2(\Omega_{\epsilon})} \leq C$ for $i \in \{1,2\}$ and similar assumptions on the forcing term, we show global existence of strong solutions; here $u^i_0$ denotes the $i$-th component of the initial data $u_0$ and $M$ is the average in the vertical direction, that is, $Mu^i_0(x_1,x_2) = (\epsilon g)^{-1} \int_{0}^{\epsilon g} u^i_0(x_1,x_2,x_3) \mathrm{d}x_3$. Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as $\epsilon \to 0$, 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 2007 text pdf 10.1512/iumj.2007.56.2834 10.1512/iumj.2007.56.2834 en Indiana Univ. Math. J. 56 (2007) 1083 - 1156 state-of-the-art mathematics http://iumj.org/access/