Measure Attractors for Stochasticnavier { Stokes Equationsmarek Capi

We show existence of measure attractors for 2-D stochas-tic Navier-Stokes equations with general multiplicative noise. Abstract. We show existence of measure attractors for 2-D sto-chastic Navier-Stokes equations with general multiplicative noise. 1. Introduction This paper is concerned with existence of attractors in connection with stochastic Navier-Stokes equations in dimension 2. For determin-istic Navier-Stokes equations, the existence of a global attractor in dimension 2 goes back to the work of Ladyzhenskaya 9] and Foias & Temam; for a full exposition see Chapter III (sec. 2) of Temam's book 15]. The new diiculties encountered when seeking attractors for the sto-chastic equations are twofold. First there is a problem with the very deenition of attractors for stochastic equations-see the discussion below. Second, for the stochastic Navier{Stokes equations there is the issue of existence of solutions to the equations themselves. Whereas the deterministic equations were solved by Leray in 1933-4 (see 14] for a modern exposition), solutions for stochastic equations with a general form of noise were rst constructed in 1991 3] some eighteen years after the rst results in this direction 2], which considered additive noise only. The notion of attractor is concerned with the asymptotic behaviour of trajectories of semigroups of operators. Recall that a semigroup on