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

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