ON THE CONSTANTS IN A KATO INEQUALITY FOR THE EULER AND NAVIER-STOKES EQUATIONS

We continue an analysis, started in [10], 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 center dot partial derivative w, 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(nd) G(n) in the Kato inequality vertical bar < v center dot partial derivative w vertical bar w > n vertical bar <= G(n) parallel to v parallel to n parallel to w parallel to(2)(n), where n epsilon (d/2 + 1, +infinity) and v, w are in the Sobolev spaces H-Sigma 0(n),H-Sigma 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. When combined with the results of [10] 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.