THE EULER EQUATIONS WITH DISSIPATION

Steady-state and time-dependent problems are studied for the equation where , is a two-dimensional closed manifold, and 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 attractors, an integer, is proved. A -weak attractor is constructed in the phase space for the velocity vortex equation.