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^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 in [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 [15][16]. 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 [3]. 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 [2]: 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.