Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"

This article offers a modern perspective which exposes the many contributions of Leray in his celebrated work on the Navier--Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth, or belongs to $H^1$, $L^2\cap L^p$ (with $p\in(3,\infty]$), as well as lower bounds on the norms $\| \nabla u (t) \|_2$, $\| u(t) \|_p$ ($p\in(3,\infty]$) as $t$ approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most $\tfrac{1}{2}$. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.Comment: 81 pages. All comments are welcom

A Bioeconomic Analysis of the Northwestern Hawaiian Islands Lobster Fishery

Several surplus production-based bioeconomic models are applied to the Northwestern Hawaiian Islands (NWHI) commercial lobster fishery. The model which best explains the biological dynamics of the fishery is a modification of the Fox model developed by the authors. Economic costs are applied within a number of conceptual frameworks to develop the first integrated bioeconomic model of the fishery. In another development, the opportunity cost of labor based on crew share at the open access equilibrium level of fishing effort is used instead of proxy wage levels. Given the costs incurred, the fishery appears to be self-regulating in terms of long-term fishing effort for maximum sustainable yield.biological production models, fisheries economics, fisheries management, spiny lobster, slipper lobster, Environmental Economics and Policy, Resource /Energy Economics and Policy,

Different kinds of long-term variability from Cygnus X-1

We present a study of the long-term variability of Cyg X-1 using data from the RXTE/ASM and the RXTE/PCA during the time between the two soft states of 1996 and 2001/2002. This period has been characterized by many short ASM flaring episodes which we have identified as "failed state transitions". The 150 d period which has been seen before and shortly after the 1996 soft state is not obviously present in the ASM rate during most of this time. Applying selection criteria from our pointed RXTE/PCA observations to exclude the flaring episodes we show that the 150 d period can indeed still be significantly detected in the hard state. Furthermore, while the ~420 d timescale associated with the flaring is reduced in the selected hard state count rate, it is still pronounced in the temporal evolution of the corresponding hardness ratios. The Ryle radio flux is also consistent with the 150 d period being present but distorted during this time.Comment: 4 pages, 6 figures, to appear in Proceedings of "X-ray Timing 2003: Rossi and Beyond", ed. P. Kaaret, F.K. Lamb, & J.H. Swan