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

We consider the incompressible Euler or Navier–Stokes (NS) equations on a torus Td, in the functional setting of the Sobolev spaces HΣ0n(Td) of divergence free, zero mean vector fields on Td, for n∈(d/2+1,+∞)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 Tc on the time of existence of the exact solution uu analyzing a posteriori   any approximate solution ua, and also to construct a function ℛnℛn such that ‖u(t)−ua(t)‖n⩽ℛn(t) for all t∈[0,Tc). Both Tc and ℛnℛn 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 (Morosi and Pizzocchero (2010, in press)  [7] and [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.