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

We consider the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w : T^d -> R^d into v . grad 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 two inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants K_{n d} \equiv K_n in the basic inequality || L(v . grad w) ||_n<= K_n || v ||_n || w ||_{n+1}, where n in (d/2, + infinity) 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 K_n, making reference to other works on the approximate solutions of Euler or NS equations.