The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

A new class of glycomimetic drugs to prevent free fatty acid-induced endothelial dysfunction

Background: Carbohydrates play a major role in cell signaling in many biological processes. We have developed a set of glycomimetic drugs that mimic the structure of carbohydrates and represent a novel source of therapeutics for endothelial dysfunction, a key initiating factor in cardiovascular complications. Purpose: Our objective was to determine the protective effects of small molecule glycomimetics against free fatty acid­induced endothelial dysfunction, focusing on nitric oxide (NO) and oxidative stress pathways. Methods: Four glycomimetics were synthesized by the stepwise transformation of 2,5­dihydroxybenzoic acid to a range of 2,5­substituted benzoic acid derivatives, incorporating the key sulfate groups to mimic the interactions of heparan sulfate. Endothelial function was assessed using acetylcholine­induced, endotheliumdependent relaxation in mouse thoracic aortic rings using wire myography. Human umbilical vein endothelial cell (HUVEC) behavior was evaluated in the presence or absence of the free fatty acid, palmitate, with or without glycomimetics (1µM). DAF­2 and H2DCF­DA assays were used to determine nitric oxide (NO) and reactive oxygen species (ROS) production, respectively. Lipid peroxidation colorimetric and antioxidant enzyme activity assays were also carried out. RT­PCR and western blotting were utilized to measure Akt, eNOS, Nrf­2, NQO­1 and HO­1 expression. Results: Ex vivo endothelium­dependent relaxation was significantly improved by the glycomimetics under palmitate­induced oxidative stress. In vitro studies showed that the glycomimetics protected HUVECs against the palmitate­induced oxidative stress and enhanced NO production. We demonstrate that the protective effects of pre­incubation with glycomimetics occurred via upregulation of Akt/eNOS signaling, activation of the Nrf2/ARE pathway, and suppression of ROS­induced lipid peroxidation. Conclusion: We have developed a novel set of small molecule glycomimetics that protect against free fatty acidinduced endothelial dysfunction and thus, represent a new category of therapeutic drugs to target endothelial damage, the first line of defense against cardiovascular disease