Strong completeness for a class of stochastic differential equations with irregular coefficients

We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded. Moreover, for each $t$, the solution flow $F_t$ is weakly differentiable and for each $p>0$ there is a positive number $T(p)$ such that for all $t<T(p)$, the solution flow $F_t(\cdot)$ belongs to the Sobolev space W_{\loc}^{1,p}. The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained