Navier-Stokes equation and forward-backward stochastic differential system in the Besov spaces

The Navier-Stokes equation on Rd (d greater or equal to 3) formulated on Besov spaces is considered. Using a stochastic forward-backward differential system, the local existence of a unique solution in B_ r, with r > 1 + d is obtained. We also show p,p p the convergence to solutions of the Euler equation when the viscosity tends to zero. Moreover, we prove the local existence of a unique solution in B_ pr,q, with p > 1, 1 greater or equal to q greater or equal to infinity, r > max(1, d); here the maximal time interval depends on p the viscosity.