PARTLY DISSIPATIVE SEMIGROUPS GENERATED BY THE NAVIER-STOKES SYSTEM ON 2-DIMENSIONAL MANIFOLDS, AND THEIR ATTRACTORS

The Navier-Stokes equations partial derivative (t)u+del(u)u=-delp+nuDE LTAu+f, div u=0, are considered on a two-dimensional compact manifold M; the phase space is not assumed to be orthogonal to the finite-dimen sional space H of harmonic vector fields on M, H = {u is-an-element-of C(infinity)(TM), DELTAu = 0}, n = dim H is the first Betti number. It is proved that the Hausdorff (and fractal) dimensions of a global att ractor A of this system satisfy dim(H)A less-than-or-equal-to c1G'2/3( 1 + ln G')1/3 + n + 1 (dim(F)A less-than-or-equal-of c2G'2/3 (1 + In G ')1/3 + 2n + 2), where G' is a number analogous to the Grashof number. In the most important particular case M = S2 (the unit sphere) the ex plicit values of the constants in the corresponding integral inequalit ies on the sphere are given, leading to the estimates dim(H)A(S2) less -than-or-equal-to 5.6G2/3 (4.3 + 4/3 ln G)1/3 + 1, diM(F)A(S2) less-th an-or-equal-to 15.8G2/3(4.3 + 4/3 ln G)1/3+2. Analogous estimates are proved for the two-dimensional Navier-Stokes equations in a bounded do main with a boundary condition that ensures the absence of a boundary layer.