Existence and continuity of exponential attractors of the three dimensional Navier-Stokes-$\alpha$ equations for uniformly rotating geophysical fluids

Abstract

Three dimensional (3D) Navier-Stokes-$\alpha$ equations are considered for uniformly rotating geophysical fluid flows (large Coriolis parameter $f=2\Omega$). The Navier-Stokes-$\alpha$ equations are a nonlinear dispersive regularization of usual Navier-Stokes equations obtained by Lagrangian averaging. The focus is on the existence and global regularity of solutions of the 3D rotating Navier-Stokes-$\alpha$ equations and the uniform convergence of these solutions to those of the original 3D rotating Navier-Stokes equations for large Coriolis parameters $f$ as $\alpha\rightarrow 0$. Methods are based on fast singular oscillating limits and results are obtained for periodic boundary conditions for all domain aspect ratios, including the case of three wave resonances which yields nonlinear resonant limit $\alpha$-equations for $f\rightarrow\infty$. The existence and global regularity of solutions of resonant limit $\alpha$-equations is established, uniformly in $\alpha$. Bootstrapping from global regularity of the resonant limit $\alpha$-equations, the existence of a regular solution of the full 3D rotating Navier-Stokes-$\alpha$ equations for large $f$ for an infinite time is established. Then we prove the existence of exponential attractors of the 3D rotating Navier-Stokes-$\alpha$ equations ($\alpha\neq 0$) and the convergence of the exponential attractors to those of the original 3D rotating Navier-Stokes equations ($\alpha =0$) for $f$ large but fixed as $\alpha\rightarrow 0$. All the estimates are uniform in $\alpha$, in contrast with previous estimates in the literature which blow up as $\alpha\rightarrow 0$.