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

We consider the incompressible Euler or Navier-Stokes (NS) equations on a torus T-d, in the functional setting of the Sobolev spaces H-Sigma 0(n)(T-d) of divergence free, zero mean vector fields on T-d, for n is an element of (d/2+1,+infinity). 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 parallel to u(t) -u(a)(t)parallel to(n) <= R-n(t) for all t is an element of [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 (Morosi and Pizzocchero (2010, in press) [7,8]). 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) [1]. 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) [9]: 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. (C) 2011 Elsevier Ltd. All rights reserved.