  In this paper we first obtain local contraction results in a Hm-norm with
respect to time and space for a local scheme. We show that a global controlled
scheme preserves higher order regularity with respect to the spatial variables
together with polynomial decay of order m of the data at each time step. Here,
we simplify the controlled scheme considered in [1] and [2]. Especially, for
the simplified controlled scheme we get an upper bound for the Leray projection
term. Furthermore, the schemes discussed in [1] and [2] are simplified in the
sense that the estimates are achieved without the use of some properties
concerning the adjoint of a local fundamental solutions with variable drift
terms. We note that the pointwise and absolute convergence of the local
functional series and their first order time derivatives and their spatial
derivatives leads to a constructive approach of local and global classical
solutions.
