  Inspired by an approach proposed previously for the incompressible
Navier-Stokes (NS) equations, we present a general framework for the a
posteriori analysis of the equations of incompressible magnetohydrodynamics
(MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev
space of infinite order, made of C^infinity vector fields (with vanishing
divergence and mean) on the torus. Given any approximate solution of the MHD
Cauchy problem, its a posteriori analysis with the method of the present work
allows to infer a lower bound on the time of existence of the exact solution,
and to bound from above the Sobolev distance of any order between the exact and
the approximate solution. In certain cases the above mentioned lower bound on
the time of existence is found to be infinite, so one infers the global
existence of the exact MHD solution. We present some applications of this
general scheme; the most sophisticated one lives in dimension d=3, with the ABC
flow (perturbed magnetically) as an initial datum, and uses for the Cauchy
problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the
conclusions arising in this case from the a posteriori analysis of the Galerkin
approximant; these include the derivation of global existence of the exact MHD
solution with the ABC datum, when the dimensionless viscosity and resistivity
are equal and stay above an explicitly given threshold value.
