On the constants in a basic inequality for the Euler and Navier-Stokes equations

We consider the incompressible Euler or Navier-Stokes equations on a d-dimensional torus ; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w into v . w’, and also involves the Leray projection L onto the space of divergence free vector fields. We derive upper and lower bounds for the constants in some inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants Kn in the basic inequality |L(v . w’) |_n <= Kn | v|_n | w |_{n+1}, where n > d/2 and v, w are in the Sobolev spaces H{n}, H{n+1} of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. Some practical motivations are indicated for an accurate analysis of the constants Kn, making reference to other works on the approximate solutions of Euler or Navier-Stokes equations.