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 . Dw, 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 K_{n d} ≡ K_n in the basic inequality ||L(v . Dw)||_n <= K_n || 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 K_n.