On approximate solutions of the incompressible Euler and Navier-Stokes equations

We consider the incompressible Euler or Navier–Stokes (NS) equations on a d-dimensional torus , in the functional setting of the Sobolev spaces H^n of divergence free, zero mean vector fields on the torus, 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 on the time of existence of the exact solution u analyzing a posteriori any approximate solution ua, and also to construct a function R_n such that |u(t) − ua(t)|_n <=R_n(t) for all t <T. Both T and R_n are determined solving suitable ‘‘control inequalities’’, depending on the error of ua; the fully quantitative implementation of this scheme depends on some previous estimates of ours on the Euler/NS quadratic nonlinearity. 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 Chernyshenko et al. (2007) . 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 Behr et al. (2001): 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.