  We consider the incompressible Euler or Navier-Stokes (NS) equations on a
torus T^d in the functional setting of the Sobolev spaces H^n(T^d) of
divergence free, zero mean vector fields on T^d, for n > d/2+1. We present a
general theory of approximate solutions for the Euler/NS Cauchy problem; this
allows to infer a lower bound T_c on the time of existence of the exact
solution u analyzing a posteriori any approximate solution u_a, and also to
construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for all t in
[0,T_c). Both T_c and R_n are determined solving suitable "control
inequalities", depending on the error of u_a; the fully quantitative
implementation of this scheme depends on some previous estimates of ours on the
Euler/NS quadratic nonlinearity [15][16]. To keep in touch with the existing
literature on the subject, our results are compared with a setting for
approximate Euler/NS solutions proposed in [3]. As a first application of the
present framework, we consider the Galerkin approximate solutions of the
Euler/NS Cauchy problem, with a specific initial datum considered in [2]: in
this case our methods allow, amongst else, to prove global existence for the NS
Cauchy problem when the viscosity is above an explicitly given bound.
