Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations

The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $\alpha>0$ and Ekman damping coefficient $\gamma>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $\alpha\to0$ and $\gamma\to0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.