CONTINUITY PROPERTIES AND GLOBAL ATTRACTORS OF GENERALIZED SEMIFLOWS AND THE NAVIER-STOKES EQUATIONS

A class of semiflows having possibly nonunique solutions is defined. T he measurability and continuity properties of such generalized semiflo ws are studied. It is shown that a generalized semiflow has a global a ttractor if and only if it is pointwise dissipative and asymptotically compact. The structure of the global attractor in the presence of a L yapunov function, and its connectedness and stability properties are s tudied. In particular, examples are given in which the global attracto r is a single point but is not Lyapunov stable. The existence of a glo bal attractor for the 3D incompressible Navier-Stokes equations is est ablished under the (unproved) hypothesis that all weak solutions are c ontinuous from (0, infinity) to L-2.