A variational approach to the Navier-Stokes equations

We propose a time discretization of the Navier-Stokes equations inspired by the theory of gradient flows. This discretization produces Leray/Hopf solutions in any dimension and suitable solutions in dimension 3. We also show that in dimension 3 and for initial datum in H 1, the scheme converges to strong solutions in some interval [0, T) and, if the datum satisfies the classical smallness condition, it produces the smooth solution in [0, ∞). © 2012 Elsevier Masson SAS