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^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, arXiv:1007.4412v2 [mathAP]; Morosi and Pizzocchero, CPAA, 2012]. 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., J. Math. Phys., 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., M2AN: Math. Model. Numer. Anal., 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.