2 research outputs found
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.Comment: LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412
by the same authors, not concerning the main result
Mentoring and the Dynamics of Affirmative Action
We analyze the long-term workforce composition when the quality of mentoring available to majority and minority juniors depends on their representation in the workforce. A workforce with ≥ 50% majority workers invariably converges to one where the majority is overrepresented relative to the population. To maximize welfare, persistent interventions, such as group-specific fellowships, are often needed, and the optimal workforce may include minority workers of lower innate talent than the marginal majority worker. We discuss the role of mentorship determinants, talent dispersion, the scope of short-term interventions, various policy instruments and contrast our results to the classic fairness narrative