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

Abstract

We consider the incompressible Euler or Navier–Stokes (NS) equations on a torus T d , in the functional setting of the Sobolev spaces H Σ 0 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 ∈ [ 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.