# Diffusions with Rough Drifts and Navier-Stokes Equation

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According to DiPerna-Lions theory, velocity fields with weak

derivatives in L^ p spaces possess weakly regular flows. When a velocity

field is perturbed by a white noise, the corresponding (stochastic) flow

is far more regular in spatial variables; a diffusion with a drift in a

suitable L p space possesses weak derivatives with exponential bounds.

As an application we show that a Hamiltonian system that is perturbed by a

white noise produces a symplectic flow for a Hamiltonian function that is

merely in W^{ 1,p} for p strictly larger than dimension. I also discuss

the potential application of such regularity bounds to study solutions of

Navier-Stokes equation with the aid of Constantin-Iyer's circulation

formula.