Hydrostatic Stokes Equations With Non-smooth Date For Mixed Boundary Conditions

The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions. In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem. As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001)]. © 2004 Elsevier SAS. All rights reserved.216807826Amrouche, C., Girault, V., Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension (1994) Czechoslovak Math. J., 44 (119), pp. 109-140Azérad, P., Guillén, F., Mathematical justification of the hydrostatic approximation in the Primitive Equations of qeophysical fluid dynamics (2001) SIAM J. Math. Anal., 33 (4), pp. 847-859Besson, O., Laydi, M.R., Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation (1992) M2AN-Mod. Math. Ana. Nume., 7, pp. 855-865Cattabriga, L., Sur un problema al contorno relativo al sistema di equazioni di Stokes (1961) Rend. Mat. Sem. Univ. Padova, 31, pp. 308-340Chacón, T., Guillén, F., An intrinsic analysis of existence of solutions for the hydrostatic approximation of the Navier-Stokes equations (2000) C. R. Acad. Sci. Paris, Série I, 330, pp. 841-846Conca, C., Stokes equations with non-smooth data (1989) Revista de Matemáticas Aplicadas, 10, pp. 115-122Girault, V., Raviart, P.A., (1986) Finite Element Methods for Navier-Stokes Equations, , Berlin: Springer-VerlagGuillén-González, F., Rodríguez-Bellido, M.A., On the strong solutions of the Primitive Equations in 2D domains (2002) Nonlin. Anal., 50, pp. 621-646Guillén-González, F., Masmoudi, N., Rodríguez-Bellido, M.A., Anisotropic estimates and strong solutions of the Primitive Equations (2001) Differential Integral Equations, 14 (11), pp. 1381-1408Lewandowski, R., (1997) Analyse Mathématique et Océanographie, , MassonLions, J.L., Magenes, E., (1969) Problèmes aux Limites Non Homogènes et Applications, 1. , Paris: DunodLions, J.L., Temam, R., Wang, S., New formulation of the primitive equations of the atmosphere and applications (1992) Nonlinearity, 5, pp. 237-288Lions, J.L., Temam, R., Wang, S., On the equations of the large scale ocean (1992) Nonlinearity, 5, pp. 1007-1053Pedlosky, J., (1987) Geophysical Fluid Dynamics, , Berlin: Springer-VerlagTemam, R., (1977) Navier-Stokes Equations: Theory and Numerical Analysis, , Amsterdam: North HollandZiane, M., Regularity results for Stokes type systems (1995) Appl. Anal., 58, pp. 263-29

Risk management considerations for cost-effective environmental decisionmaking

Scientific publications and media reports continually remind us about the environmental hazards that surround us. We are appraised of the environmental legacies left by chemical industries, the defense complex, and even our local dry cleaning establishments. Governmental regulations have dictated that industry provide detailed listings of their input materials, wastes, and emissions to the public and perform risk assessments to demonstrate compliance with standards. These regulations were designed to make industry more accountable and to give the public information that would allow them to understand risks and either work for change or accept their living conditions. This process would appear to be rational, fair, and acceptable to both industry and the public. However, our inability to reach agreement on questions such as ``How Clean is Clean?`` or ``Is It Safe?`` after more than ten years of scientific and public discussions, coupled with the frequency of environmental demonstrations throughout the world, serves as evidence that ``acceptable risk`` has not yet been defined

A traditional mediterranean diet effectively reduces inflammation and improves cardiovascular health

Chrysohoou et al. fifteen years ago, showed in an elegant analysis nested within the ATTICA study [1] that a dietary score reflecting adherence to the traditional Mediterranean diet (MedDiet) was inversely associated with plasma biomarkers of low-grade inflammation. Specifically, participants in the highest tertile of adherence to the MedDiet presented 20% lower levels of highly-sensitive C reactive protein (hs-CRP), 17% lower levels of interleukin-6 (IL-6), and 14% lower white blood cell counts. This was an observational study that could be affected by residual confounding and other potential imperfections. However, another similar study, this time nested within the Nurses Cohort in the USA [2], assessed hs-CRP, IL-6, E-selectin, soluble intercellular cell adhesion molecule 1, and soluble vascular cell adhesion molecule 1 and found that better adherence to the MedDiet was also associated with a reduction in inflammatory biomarker concentrations, with relative reductions of 24% in hs-CRP, 16% in IL-6, and 13% in E-selectin concentrations [2]. These well conducted observational studies were subsequently confirmed by a randomized clinical trial (the pilot study of the PREDIMED (PREvención con DIeta MEDiterránea) trial) where we were able to show that an intervention with 2 MedDiets maintained during 3 months was able to reduce hs-CRP, IL-6 (in both cases) and adhesion molecules compared to a low-fat diet [3]. However, hs-CRP was reduced only when the MedDiet was supplemented with polyphenol-rich extra-virgin olive oil, but not with nuts

Mediterranean diet as the ideal model for preventing non-alcoholic fatty liver disease (NAFLD)

The prevalence of nonalcoholic fatty liver disease (NAFLD) has dramatically increased in the 2–3 last decades and it represents the most frequent global cause of liver disease, affecting 25% to 45% of adults in most studies (1). The worldwide elevation in the population rates of NAFLD has come in parallel with rising unprecedented pandemics of obesity and diabetes. There is no specific medication for NAFLD and dietary/lifestyle modifications are the main foundations for the treatment of NAFLD. They are also very likely to be effective for its primary prevention. In this context, a dietary pattern that meets most requirements to become the ideal model for the prevention of NAFLD is the traditional Mediterranean diet (2). In addition to the potential for preventing NAFLD, the Mediterranean diet has sufficiently demonstrated its great ability to improve cardiovascular health. The accrual of prospective and well conducted studies showing cardiovascular benefits associated with better adherence to a high-quality food pattern is superior for the traditional Mediterranean food pattern than for any other dietary pattern (3)

Prevención cardiovascular y de la osteoporosis con terapia hormonal sustitutiva

Se recomienda actualmente que todo médico -sea cual sea su especialidad- aproveche la oportunidad que supone la atención clínica a cualquier mujer postmenopáusica o que se halle en torno a la menopausia para aconsejarle acerca de los beneficios y riesgos que supone la terapia hormonal sustitutiva (THS) con estrógenos y progesterona. Una vez que sea adecuadamente informada de estos beneficios y riesgos, debe ser la propia mujer quien decida si va a iniciar o no esta terapia, teniendo en cuenta no sólo su propio perfil de riesgo sino también sus preferencias. Los beneficios más claros se refieren a la reducción del riesgo de osteoporosis y de fracturas, así como al control y prevención de algunos síntomas frecuentes en la menopausia. Para prevenir la osteoporosis se requiere un uso continuado de IaTHS. También existen cada vez más argumentos que sugieren una reducción del riesgo de un primer episodio coronario. No obstante, no hay pruebas de que laTHS sea útil en la prevención secundaria de la cardiopatía isquémica o aporte alguna protección frente a los accidentes cerebrovasculares. Las desventajas más importantes de la THS se refieren a un mayor riesgo de hiperplasia de endometrio y de cáncer endometrial cuando se usan estrógenos solos, sin combinarlos con progestágenos. También existe un pequeño incremento del riesgo de cáncer de mama cuando se prolonga la THS, también cuando se asocian progestágenos

Clifford algebra with mathematica

The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main ideas of Clifford algebra, with the main goal to develop a package for Clifford algebra calculations for the computer algebra program Mathematica. The Clifford algebra package is thus a powerful tool since it allows the manipulation of all Clifford mathematical objects. The package also provides a visualization tool for elements of Clifford Algebra in the 3-dimensional space