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

TL;DR

This paper derives explicit upper bounds for the fractal dimension of attractors in the 3D damped Euler-Bardina equations, analyzing how these bounds depend on regularization and damping parameters, and demonstrates their sharpness in certain limits.

Contribution

It provides the first explicit upper bounds for the attractor dimension of the 3D damped Euler-Bardina equations, including sharpness analysis as parameters approach classical Euler equations.

Findings

Derived explicit upper bounds for attractor dimension.

Showed bounds are sharp as regularization and damping parameters tend to zero.

Analyzed the dependence of attractor dimension on key parameters.

Abstract

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.