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 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. 
Carlo Morosi (Politecnico di Milano), Livio Pizzocchero (Universita'   di Milano)
math.AP math-ph math.FA math.MP