Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 . D 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_{n d} = G_n in the Kato inequality | < v
. D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity)
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 [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.

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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph
3
The categories of this paper are: physics
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci