Partly dissipative semigroups generated by the Navier–Stokes system on two-dimensional manifolds, and their attractors

The Navier–Stokes equations
$$ \partial_tu+\nabla_uu=-\nabla p+\nu\Delta u+f, \qquad \operatorname{div}u=0, $$
are considered on a two-dimensional compact manifold $M$; the phase space is not assumed to be orthogonal to the finite-dimensional space $\mathscr{H}$ of harmonic vector fields on $M$, $\mathscr H=\{u\in C^\infty(TM),\,\Delta u=0\}$, $ n=\dim\mathscr H$ is the first Betti number. It is proved that the Hausdorff (and fractal) dimensions of a global attractor $\mathscr A$ of this system satisfy $\dim_H\mathscr A\leqslant c_1G'^{2/3}(1+\ln G')^{1/3}+n+1$ $(\dim_F\mathscr A\leqslant c_2G'^{2/3}(1+\ln G')^{1/3}+2n+2)$, where $G'$ is a number analogous to the Grashof number. In the most important particular case $M=S^2$ (the unit sphere) the explicit values of the constants in the corresponding integral inequalities on the sphere are given, leading to the estimates, $\dim_H\mathscr A_{S^2}\leqslant 5.6G^{2/3}(4.3+\frac43\ln G)^{1/3}+1$, $\dim_F\mathscr A_{S^2}\leqslant 15.8G^{2/3}(4.3+\frac43\ln G)^{1/3}+2$. Analogous estimates are proved for the two-dimensional Navier–Stokes equations in a bounded domain with a boundary condition that ensures the absence of a boundary layer.