  In a previous paper of ours [Nonlinear Anal. 2012] we have considered the
incompressible Navier-Stokes (NS) equations on a d-dimensional torus T^d, in
the functional setting of the Sobolev spaces H^n(T^d) of divergence free, zero
mean vector fields (n > d/2+1). In the cited work we have presented a general
setting for the a posteriori analysis of approximate solutions of the NS Cauchy
problem; given any approximate solution u_a, this allows to infer a lower bound
T_c on the time of existence of the exact solution u and to construct a
function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for t in [0,T_c). In
certain cases it is T_c = + infinity, so global existence is granted for u. In
the present paper the framework of [Nonlinear Anal., 2012] is applied using as
an approximate solution an expansion u^N(t) = Sum_{j=0}^N R^j u_j(t), where R
is the Reynolds number. This allows, amongst else, to derive the global
existence of u when R is below some critical value R_{*} (increasing with N in
the examples that we analyze). After a general discussion about the Reynolds
expansion and its a posteriori analysis, we consider the expansions of orders
N=1,2,5 in dimension d=3, with the initial datum of Behr, Necas and Wu [M2AN,
2001]. Computations of order N=5 yield a quantitative improvement of the
results previously obtained for this initial datum in [Nonlinear Anal. 2012],
where a Galerkin approximate solution was employed in place of the Reynolds
expansion.
