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 ⋅ ∂ 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 ≡ K n in the basic inequality ‖ L ( v ⋅ ∂ w ) ‖ n ⩽ K n ‖ v ‖ n ‖ w ‖ n + 1 , where n ∈ ( d / 2 , + ∞ ) and v , w are in the Sobolev spaces H Σ 0 n , H Σ 0 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.