By Chérif Amrouche, Ulrich Razafison (auth.), Rolf Rannacher, Adélia Sequeira (eds.)

This publication is a special selection of high-level papers dedicated to primary subject matters in mathematical fluid mechanics and their purposes, generally in reference to the clinical paintings of Giovanni Paolo Galdi. The contributions are customarily founded at the examine of the fundamental houses of the Navier-Stokes equations, together with lifestyles, forte, regularity, and balance of options. comparable versions describing non-Newtonian flows, turbulence, and fluid-structure interactions also are addressed. the consequences are analytical, numerical and experimental in nature, making the booklet rather attractive to an unlimited readership encompassing mathematicians, engineers and physicists. the range of the themes, as well as different ways, will supply readers a world and updated evaluation of either the most recent findings at the topic and of the salient open questions.

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**Sample text**

33) Let now p ∈ ]1, 2[. There exist p0 and p1 such that 1 < p0 < p < p1 < 2 and such that the operator R : f −→ O ∗ f is continuous from L p0 (R3 ) 2 p0 2 p1 ,∞ ,∞ into L 2− p0 (R3 ) and from L p1 (R3 ) into L 2− p1 (R3 ). The Marcinkiewicz theorem allows again to conclude that the operator R is continuous from L p (R3 ) into 2p L 2− p (R3 ) 4 (ii) The same remark remains valid for ∇ O that belongs to L 3 ,∞ (R3 ). Using the Young inequality with the relations (10) and (11), we get the following result: Proposition 1 Let f ∈ L 1 (R3 ).

Princeton, NJ, 1970 A New Model of Diphasic Fluids in Thin Films Guy Bayada, Laurent Chupin, and B´er´enice Grec Abstract In this work, we are interested in the modelling of diphasic fluids flows in thin films. The diphasic aspect is described by a diffuse interface model, the Cahn-Hilliard equation. The specific geometry (thin domain) allows to replace heuristically the usual Navier-Stokes equations by an asymptotic approximation, a modified Reynolds equation (in which the pressure and the velocity are uncoupled), where the viscosity depends on the composition of the mixture.

It has been proved in [1] that this limit can be justified rigorously. Introducing the rescaled domain Ω = (x, y) ∈ R2 , 0 < x < L , 0 < y < h(x) , the following steady-state equation is obtained to the limit ε → 0: ∂ y η ∂ y u = ∂x p, ∂ y p = 0, ∂x u + ∂ y v = 0. (4) The usual procedure to obtain the Reynolds equation is to integrate twice (4) with respect to y, and make use of the boundary conditions (2), u can be expressed as a function of p. The incompressibility condition enables to obtain an equation on the pressure only, the Reynolds equation: ∂x h3 ∂x p = s∂x 12η h .