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

We continue an analysis, started in a previous paper of ours, of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map B(v, w), where v and w 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 Gn in the Kato inequality <B(v,w), w>_n <= Gn |v|_n |w|_n , where n > d/2 + 1, and v, w are in the Sobolev spaces 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 our previous paper 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.