Navier-Stokes and Euler equations on two-dimensional closed manifolds. (Russian) Zbl 0713.35074

The author considers Navier-Stokes equations \[ \partial_ tu+\nabla_ uu+\nu \Delta u=-\nabla p+f,\quad div u=0 \] on a two-dimensional closed manifold M, embedded in \({\mathbb{R}}^ 3\). The theorems of existence and uniqueness of generalized solutions to the stationary and nonstationary problems have been proved. The unique solvability of Euler equations \((\nu =0)\) is obtained by the vanishing viscosity method \(\nu \to +0\). The author also obtains existence of maximal attractor to Navier-Stokes equations on M, and estimates the Hausdorff dimension of the attractor \[ \dim {\mathcal A}_{S^ 2}\leq C(\nu^{-8/3}\| f\|^{4/3}+\nu^{- 2}\| f\|), \] when M is the sphere \(S^ 2\).
Reviewer: J.Wang


35Q30 Navier-Stokes equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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