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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 $alphato0$ and $gammato0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.
We prove existence of the global attractor of the damped and driven 2D Euler--Bardina equations on the torus and give an explicit two-sided estimate of its dimension that is sharp as $alphato0^+$.
We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when t
We prove existence of the global attractor of the damped and driven Euler--Bardina equations on the 2D sphere and on arbitrary domains on the sphere and give explicit estimates of its fractal dimension in terms of the physical parameters.
In this paper we give optimal lower bounds for the blow-up rate of the $dot{H}^{s}left(mathbb{T}^3right)$-norm, $frac{1}{2}<s<frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an elementary proof for a l
We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $dot H^s(R^3)$ with $frac 12 <s<frac{3}{2}$. In case $frac 12 <s leq 1$ we show that the lower bounds are sharp.