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 Td; the quadratic term in these equations arises from the bilinear map sending two velocity fields v,w:Td→Rd 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 two inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants Knd≡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 HΣ0n,HΣ0n+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 NS equations.