The Euler equations with dissipation

Steady-state and time-dependent problems are studied for the equation
$$ \partial_tu+\Pi(\nabla_uu)=-\sigma u+f, $$
Where $u\in TM$, $M$ is a two-dimensional closed manifold, and $\Pi$ is the projection onto the subspace of solenoidal vector fields that admit a single-valued flow function. Existence of steady-state solutions is proved. For the evolution problem Lyapunov stability of the zero solution in Sobolev–Liouville spaces is proved by the method of vanishing viscosity. The existence of generalized weak $(\Pi W_{2k}^1,\Pi W_{2kw}^1)$ attractors, $k\geqslant1$ an integer, is proved. A $*$-weak $(\mathring{L}_\infty,\mathring{L}_{\infty\,*\text{-}\omega})$ attractor is constructed in the phase space $\mathring{L}_\infty$ for the velocity vortex equation.