  In this paper we study properties of solutions to stochastic differential
equations with Sobolev diffusion coefficients and singular drifts. The
properties we study include stability with respect to the coefficients, weak
differentiability with respect to starting points, and the Malliavin
differentiability with respect to sample paths. We also establish
Bismut-Elworthy-Li's formula for the solutions. As an application, we use the
stochastic Lagrangian representation of incompressible Navier-Stokes equations
given by Constantin-Iyer \cite{Co-Iy} to prove the local well-posedness of NSEs
in $\mR^d$ with initial values in the first order Sobolev space
$\mW^1_p(\mR^d;\mR^d)$ provided $p>d$.
