372 research outputs found

### Global regularity of the Navier-Stokes equation on thin three dimensional domains with periodic boundary conditions

This paper gives another version of results due to Raugel and Sell, and
similar results due to Moise, Temam and Ziane, that state the following: the
solution of the Navier-Stokes equation on a thin 3 dimensional domain with
periodic boundary conditions has global regularity, as long as there is some
control on the size of the initial data and the forcing term, where the control
is larger than that obtainable via ``small data'' estimates. The approach taken
is to consider the three dimensional equation as a perturbation of the equation
when the vector field does not depend upon the coordinate in the thin
direction.Comment: Also available at http://math.missouri.edu/~stephen/preprint

### Analytic measures and Bochner measurability

Let $\Sigma$ be a $\sigma$-algebra over $\Omega$, and let $M(\Sigma)$ denote
the Banach space of complex measures. Consider a representation $T_t$ for
$t\in\Bbb R$ acting on $M(\Sigma)$. We show that under certain, very weak
hypotheses, that if for a given $\mu \in M(\Sigma)$ and all $A \in \Sigma$ the
map $t \mapsto T_t \mu(A)$ is in $H^\infty(\Bbb R)$, then it follows that the
map $t \mapsto T_t \mu$ is Bochner measurable. The proof is based upon the idea
of the Analytic Radon Nikod\'ym Property.
Straightforward applications yield a new and simpler proof of Forelli's main
result concerning analytic measures ({\it Analytic and quasi-invariant
measures}, Acta Math., {\bf 118} (1967), 33--59)

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