Abstract
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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \to {{0}<^>{ + }}$$\end{document}, where alpha is the parameter describing smoothing of the vector field in the nonlinear term.