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Abstract
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 $(v, w) \mapsto v \cdot \partial w$, where $v, w : T^d \to R^d$ 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
$G_{n d} \equiv G_n$ in the Kato inequality
$| < v \cdot \partial w | w >_n |
\leq G_n || v ||_n || w ||^2_n$, where
$n \in (d/2 + 1, + \infty)$ and
$v, w$ are in the Sobolev spaces $H^n_{\Sigma_0}, H^{n+1}_{\Sigma_0}$
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.
Mathematics Subject Classification: Primary: 76D05, 26D10; Secondary: 46E35.
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