Sharp dimension estimates for the attractors of the regularized damped Euler system

A regularized damped Euler system in two-dimensional and three-dimensional setting is considered. The existence of a global attractor is proved and explicit estimates of its fractal dimension are given. In the case of periodic boundary conditions both in two-dimensional and three-dimensional cases, it is proved that the obtained upper bounds are sharp in the limit $a\to0^+$, where $a$ is the parameter describing smoothing of the vector field in the nonlinear term.