THE EULER EQUATIONS WITH DISSIPATION

Steady-state and time-dependent problems are studied for the equation partial derivative(t)u + PI(del(u)u) = - sigmau + f, where u is-an-ele ment-of TM, M is a two-dimensional closed manifold, and PI is the proj ection onto the subspace of solenoidal vector fields that admit a sing le-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 viscosi ty. The existence of generalized weak (PIW2k1, PIW2kw1) attractors, k > 1 an integer, is proved. A -weak (L(infinity), L(infinity*-w) attra ctor is constructed in the phase space L(infinity) for the velocity vo rtex equation.