The Navier–Stokes and Euler equations on two-dimensional closed manifolds

The Navier–Stokes equations
$$ \partial_tu+\nabla_uu+\nu\Lambda u=-\nabla p+f, \qquad \operatorname{div}u=0 $$
are considered on a two-dimensional closed manifold $M$ imbedded in $R^3$. Theorems on existence and uniqueness of generalized solutions of steady-state and time-dependent problems are proved. Unique solvability of the Euler equations $(\nu=0)$ is proved by passing to the limit as $\nu\to+0$. The existence of a maximal attractor for the Navier–Stokes system on $M$ is proved, and for the case where the manifold $M$ is the sphere $S^2$ an estimate for the Hausdorff dimension of the attractor is obtained:
$$ \dim\mathscr A_{S^2}\leqslant c(\nu^{-8/3}\|f\|^{4/3}+\nu^{-2}\|f\|). $$