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

We continue an analysis, started in (C. Morosi, L. Pizzocchero, arXiv:1007.4412v2 [math.AP] (2010)), of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v,w) → v . Dw, where v,w : T^d → R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} ≡ G_n in the Kato inequality |< v . Dw | w>_n| <= G_n || v ||_n || w ||^2_n, where n ∈ (d/2 + 1,+∞) 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. When combined with the results of (C. Morosi, L. Pizzocchero, arXiv:1007.4412v2 [math.AP] (2010)) on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.