Weak vorticity formulation for the incompressible Euler equations in domains with boundary

In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data We begin by showing that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Next, we focus on weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove that the circulations around each boundary component cannot be smaller than the initial data circulation, so that nonnegative vorticity may be absorbed by the boundary, but not produced by the boundary. In addition, we prove that if the weak solution conserves circulation at the boundary components it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation at the boundary components if the initial vorticity is integrable. In addition, we discuss the definition of the mechanical force which the flow exerts on material boundary components and its relation with conservation of circulation. Finally, we describe the corresponding results for a bounded domain with holes, and the adaptations required in the proofs.Comment: 37 page

Correlation of magnetic susceptibility with 18O data in magnetite- and ilmenite-type granites from Iberian massif

The relationship between oxygen isotopic values and magnetic susceptibility composition on 11 Variscan Portuguese granites has been investigated. Whole-rock oxygen-isotope (18O ) values for Vieira do Minho (VM), Vila Pouca de Aguiar (VPA), Chaves, Castelo Branco (CB), Manteigas and Serra da Estrela (SE) granitoids, were compilated from bibliography [1,2,3,4], and 18O for Santa Eulalia Plutonic Complex (SEPC) were obtained by laser fluorination at the Stable Isotopic Laboratory of Salamanca. Magnetic susceptibility (Km) values were obtained with a Kappabridge equipment from Toulouse University and Geology Centre, Porto University [2,5,6,7,8]. In this study is shown that there is a significant inverse correlation between Km and 18O. Magnetite-type granites (Manteigas granodiorite and SEPC external facies) have Km>10-3 SI and low 18O values ranging from 8.9 to 10.3 0/00 instead those of ilmenite-type (all the other granites) have Km 10-4 SI and are 18O enriched (9.3 to 13.5 0/00). The I-type granites (VM, VPA, Chaves, Manteigas and SEPC external facies) show lower average 18O (10.2 0/00) and higher Km values (100x10-6 SI) than the S-type granites (SE and CB) with 18O = 12.6 0/00 and Km = 65x10-6 SI. Figure 1: Semi-log plot of Km (in Si units) versus 18O. This work has been financially supported by PTDC/CTEGIX/ 099447/2008 (FCT-Portugal, COMPETE/FEDER). [1] Martins et al. (in prep.) [2] Martins et al. (2009) Lithos 111, 142-155. [3] Antunes et al. (2008) Lithos 103, 445-465. [4] Neiva et al. (2009) Lithos 111, 186-202. [5] Sant’Ovaia et al. (2010) JSG 32, 1450-1465. [6] Sant’Ovaia et al. (2000) TRSE, ES 91, 123-127. [7] Sant’Ovaia et al. (2008) 33rd IGC CD. [8] Sant’Ovaia et al. (2011) Min. Mag. 75, 3, 1795

From Self-Interest Motives to Justice Motives The Challenges of Some Experimental Results

WOS:000256153600006 (Nº de Acesso Web of Science)This article begins by presenting experimental evidence that remains unexplained by standard and utility-extended economic models: experimental subjects tend to honor their promises even on occasions when an assessment of consequences asks them to defect; subjects voluntarily contribute to collective goods, and this contribution is highly conditional on others contributing as well; subjects evaluate and value the intentions behind actions as well as the consequences of actions. Arguments are sought for in moral philosophy that would more plainly explain the collected experimental evidence and that would help economists revise their explanatory frames. The hypothesis advanced is that the observed behavior may be interpreted as resulting from the moral strength of indignation and justice norms