Cet ouvrage est consacré aux espaces vectoriels normés ou semi-normés, dont les espaces de Banach, Fréchet et Hilbert, avec des développements nouveaux sur les espaces de Neumann – c’est-à-dire dans lesquels toute suite de Cauchy converge – et sur les espaces extractables – c’est-à-dire dans lesquels toute suite bornée a une sous-suite faiblement convergente.Il présente les principales propriétés de ces espaces utiles pour la construction des espaces de distributions, de Lebesgue et de Sobolev, à valeurs réelles ou vectorielles, ainsi que pour la résolution d’équations aux dérivées partielles. Dans ce but, le calcul différentiel est étendu aux espaces semi-normés.Espaces de Banach, Fréchet, Hilbert et Neumann privilégie les méthodes simples, les semi-normes, les propriétés séquentielles et bien d’autres encore, afin de rendre ces outils accessibles au plus grand nombre – doctorants, étudiants de troisième cycle, ingénieurs – sans en restreindre la généralité.

We show that the common identification of H with its dual space in the famous functional frame V ⊂ H = H′ ⊂ V′ is incompatible with the distributional frame for some standard pdes and we recall an error related to this unnecessary identification which is repeated from years.

We present a set of equations modelling wind velocity in a 3D domain in terms of the ground height function, the ground temperature and the wind on the boundary. The wind field is adjusted to several punctual wind velocity measurements at different points in the 3D domain by an optimal control problem in which the (unknown) wind on the boundary is the control. Using the meteorological wind punctual measures as datum, the model provides locally a detailed 3D wind that takes into account topography and thermal gradients on the surface by solving only 2D linear equations. We consider briefly the numerical approximation and two examples, including one with real data to control the accuracy of the model. Copyright (C) 2009 John Wiley & Sons, Ltd.

We present a convection model which can be coupled with fire propagation models in order to take into account the wind and the slope which are two of the most relevant factors affecting surface fire spread. An asymptotic analysis gives a three-dimensional convective model governed by a two-dimensional equation.

We study the ow of a uid occupying an innnite horizontal domain bounded by a rugose plate which is at rest and by a plane plate which moves with a constant velocity. The rugose plate is covered with periodically distibuted asperities of size ". We prove that, outside a neighbourhood of the rugose region, the ow behaves asymptotically as a Couette ow, as " ! 0, up to an exponentially small error. Innuence de la rugosit e sur un ecoulement gouvern e par les equations de Navier{Stokes R esum e. On etudie l' ecoulement d'un uide connn e entre une plaque rugueuse qui est au repos et une plaque plane qui se d eplace a vitesse constante. La plaque rugueuse est couverte d'asp erit es de taille " r eparties p eriodiquement. On montre qu'en dehors d'un voisinage des asp erit es, l' ecoulement se comporte asymptotiquement comme un ecoulement de Couette, quand " ! 0, avec une erreur exponentiellement petite. Abstract. We study the ow of a uid occupying an innnite horizontal domain bounded by a rugose plate which is at rest and by a plane plate which moves with a constant velocity. The rugose plate is covered with periodically distibuted asperities of size ". We prove that, outside a neighbourhood of the rugose region, the ow behaves asymptotically as a Couette ow, as " ! 0, up to an exponentially small error.

The main purpose of this paper is to justify rigorously the following assertion: A viscous fluid cannot slip on a wall covered by microscopic asperities because, due to the viscous dissipation, the surface irregularities bring to rest the fluid particles in contact with the wall. In mathematical terms, this corresponds to an asymptotic property established in this paper for any family of fields that slip on oscillating boundaries and remain uniformly bounded in the H1-norm.

We prove, on one hand, that for a convenient body force with valuesin the distribution space (H -1(D)) d , where D is the geometricdomain of the fluid, there exist a velocity u and a pressure psolution of the stochastic Navier–Stokes equation in dimension2, 3 or 4. On the other hand, we prove that, for a body force with values in thedual space V’ of the divergence free subspace V of (H 1 0(D)) d ,in general it is not possible to solve the stochastic Navier–Stokesequations. More precisely, although such body forces have been considered, thereis no topological space in which Navier–Stokes equations could bemeaningful for them.

In the case of a constant depth, western intensification of currents in oceanic basins was mathematically recovered in various models (such as Stommel, Munk or quasi-geostrophic ones) as a boundary layer appearing when the solution of equations converges to the solution of a pure transport equation. This convergence is linked to the fact that any characteristic line of the transport vector field included in the equations crosses the boundary, and the boundary layer is located at outgoing points.Here we recover such a boundary layer for the vertical-geostrophic model with a general bathymetry. More precisely, we allow depth to vanish on the shore in which case the above mentioned characteristic lines no longer cross the boundary. However a boundary layer still appears because the transport vector field a (which is tangential to the boundary) locally converges to a vector field (a) over bar with characteristic lines crossing the boundary.

We present a model coupling the fire propagation equations in a bidimensional domain representing the surface, and the air movement equations in a three dimensional domain representing an air layer. As the air layer thickness is small compared with its length, an asymptotic analysis gives a three dimensional convective model governed by a bidimensional equation verified by a stream function. We also present the numerical simulations of these equations.

Phthalocyanine and lutetium bisphthalocyanine derivatives functionalized with crown - ether moieties and a carboxylic eater terminated side chain, are synthesized and characterized. The complexation properties towards potassium ion are studied by UV - visible absorption spectrometry. A procedure is defined to quantitatively determine the positive cooperative effect arising during the complexation processes.

We study the effect of the rugosity of a wall on the solution of the Stokes system complemented with Fourier boundary conditions. We consider the case of small periodic asperities of size ε. We prove that the velocity field, pressure and drag, respectively, converge to the velocity field, pressure and drag of a homogenized Stokes problem, where a different friction coefficient appears. This shows that, contrarily to the case of Dirichlet boundary conditions, rugosity is dominant here. Copyright © 2001 John Wiley & Sons, Ltd.

Starting from 1,2-naphthalocyanine lutetium acetate (1,2- NcLuOAc ) the unsymmetrical lutetium phthalo(naphthalo)cyanine (1,2- NcLuPc ) has been synthesized and characterized by optical absorption spectra and fast atom bombardment (FAB) mass spectrometry. 1,2-Naphthalocyanine shows several geometrical isomers, both the mixture (noted Nc ∑ ) and one of them ( Nc Cs ) have been used for forming the corresponding bismacrocycles. As a result the first racemic mixture of a chiral unsubstituted phthalo(naphthalo)cyanine complex has been synthesized and characterized.

We investigate the steady motion of a liquid in a lake, modeled as a thin domain. We assume the motion is governed by Navier—Stokes equations, while a Robin-type traction condition, and a friction condition is prescribed at the surface and at the bottom, respectively. We also take into account Coriolis forces. We derive an asymptotic model as the aspect ratio \( \delta \) = depth/width of the domain goes to 0. When the Reynolds number is not too large, this is mathematically justified and the three-dimensional limit velocity is given in terms of wind, bathymetry, depth and of a two-dimensional potential. Numerical simulation is carried out and the influence of traction condition reading is experienced.

We study the stationary flow of a fluid occupying a 3D infinite horizontal domain bounded by a rough wall that is at rest and by a plane that moves with a constant velocity. The rough wall is a plane covered with periodically distributed asperities of size ε \varepsilon . We prove that, outside a neighbourhood of the rough region, the flow behaves asymptotically as a Couette flow, as ε → 0 \varepsilon \to 0 , up to an exponentially small error.

The motion of a fluid subject to Navier–Stokes equations with Coriolis force, to a traction condition at the surface and to a friction condition at the bottom is investigated. An asymptotic model is derived as the aspect ratio δ = depth/width of the domain goes to 0. The 3D limit velocity is given in terms of wind, bathymetry, depth and of a 2D potential. Numerical simulation is carried out on North Pacific.

The effect of tiny asperities covering a wall on a flow governed by Stakes equations with Fourier boundary conditions is investigated We calculate the limit flow and we give estimates of the deviations of rite drag, velocity field and pressure, in terms of the size epsilon of the asperities. In the particular case of a plate, the limit drag is larger than the drag of the smooth wall, in contrast with the situation found for Dirichlet boundary conditions. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.