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)