Weak Solutions of Non Coercive Stochastic Navier-stokes Equations in R

We prove existence of weak solutions of stochastic Navier-Stokes equations in R which do not satisfy the coercivity condition. The equations are formally derived from the critical point of some variational problem defined on the space of volume preserving diffeomorphisms in R. Since the domain of our equation is unbounded, it is more difficult to get tightness of approximating sequences of solutions in comparison with the case of a bounded domain. Our approach is based on uniform a priori estimates on the enstrophy of weak solutions of the stochastic 2D-Navier-Stokes equations with periodic boundary conditions, where the periodicity is growing to infinity combined with a suitable spatial cutoff-technique.