This project is articulated around to directions: semi-classical limit, and long time behavior, for nonlinear Schrödinfer equations. The first direction is subdivided into supercritical WKB regime (need of a new model beyond singularities for the Euler equation), critical WKB regime (and link with the Cauchy problem on the one hand, and Physics on the other hand), the propagation of wave packets, and instabilities. The second direction is concerned with dynamical properties of the nonlinear scattering operator, the role of geometry, vortex filaments, and solitons for the Gross-Pitaevskii equation.

The main objective of the proposal is to design and produce porous membranes having non-fluoro super-hydrophobic surface. To achieve this goal we propose a strategy bio-inspired from the lotus leaf effect. The surface of lotus leaves displays a hierarchical morphology based on the combination of nano- and micro-structures. The roughness developed by this unique feature gives rise to high water-repellency and self-cleaning properties without any fluorinated compounds. The purpose is to reach the control of the surface energy by mastering the fabrication process to obtain such hierarchical structures. The design and producing capability of porous membranes having controlled surface hierarchical morphology require the in-depth understanding of interplay between mass transfer operating during the processing and structure, surface properties and permeation functionalities. Surface hierarchical structures will be generated according to two different approaches: (i) during the fabrication process using vapour induced phase separation (VIPS) and (ii) by post-treatment of pre-formed porous membranes using plasma techniques. In the former method, VIPS will be used to generate the micro-scale structure and polymer crystallization and block copolymer additives for the nano-scale structure, forming the hierarchical structure. In the latter method, plasma etching and anarchical growth under unusual PECVD conditions will be investigated. Feasibility of both methods to produce the desired structure in a spatially controlled uniform surface area and with stable surface properties will be evaluated for a scaling-up capability. The novelty of the proposal lies in association between hierarchical micro- and nano- binary organisation and membrane material having a specified porosity controlled by the fabrication process. This approach is completely novel and the prepared membrane new since no report describes such fluorine-less material till now. Development of the above-mentioned fabrication processes will be carried out by using some recent methods described to turn surface super-hydrophobic. Another novelty concerns the understanding of mechanisms controlling the formation of such micro- nano- assembly to monitor uniformity and defect generation by the fabrication process. The understanding of structure-processing-property relationship is required to address questions about: (i) the spatial placement and length scale distribution of hierarchical structures and (ii) the tolerance limit for defects. This point is especially critical to envision the scaling-up stage. Beyond this proposal, the long term ambition of this project is to develop a fundamental understanding of the morphology control of membrane at multi-scaled length from nano to micro to macro. The aim is to design and manufacture membrane materials with the exact properties needed. This requires the development and validation of coupled modelling mass transfer-structure formation during the membrane processing enabling the prediction of membrane morphology.

Today, our energy needs are in large part satisfied by fossil fuels. In the future, renewable energies will play an ever increasing role. For transportation, a renewable fuel will be needed, except for battery vehicles with limited driving range. Hydrogen produced from water and renewable energies could be that fuel. In this respect, H2/air polymer electrolyte fuel cells (PEFCs) are interesting due to their twice higher energy efficiency compared to a H2-combustion engine. A major drawback of today’s PEFCs is their dependence on platinum, a rare and expensive metal, for catalyzing the PEFC-reactions: hydrogen-oxidation and air-reduction. The latter reaction is by far the slowest and 90 % of the platinum in such a fuel cell is used at the air-reducing cathodes. Based on today’s price for platinum, studies have shown that 40-50% of the material’s cost of a PEFC-stack would be ascribed to the raw platinum metal. Therefore, eliminating platinum from the cathode would drastically reduce the cost of PEFCs and allow a massive utilization of this technology. Recently, several breakthroughs have been reported in the field of non-precious-metal catalysts (NPMCs) made from iron (cobalt), nitrogen and carbon. Their activity for the oxygen reduction has been increased tremendously, making them suddenly interesting catalysts from a performance standpoint. Other less active NPMC catalysts have been reported to be stable for 700 h. In order to bring these NPMCs into real PEFC stacks, the highest activity reported for recent NPMCs will have to be combined with a stable behaviour for thousands of hours in an operating PEFC, as required for transportation application. The aim of the present proposal is to investigate innovative approaches to obtain more durable NPMCs and simultaneously advance the science on the various degradation mechanisms that are specific to these catalysts in fuel-cell environment. Two novel approaches will be investigated. The first will consist in synthesizing new NPMCs by replacing the microporous carbon support by other microporous supports. The second will consist in modifying the surface of pre-existing NPMCs by various methods, in order to strengthen the resistance of Fe-based catalytic sites to demetallation, oxidative attack or anion adsorption. Simultaneously, an experimental methodology will be developed to quantify the importance and rate of each of these degradation mechanisms. Long fuel cell tests under controlled conditions will be coupled with advanced characterization techniques such as X-ray photoelectron spectroscopy, Mössbauer spectroscopy and on-line mass spectroscopy.

In some classes of reaction diffusion equations, solutions may develop sharp internal layers, or interfaces, that separate the spatial domain into different phase regions. This happens, in particular, when the reaction term is very large compared with the diffusion term.This project is concerned with the singular limit of reaction diffusion equations, as a parameter related to the thickness of a diffuse interface tends to zero. We shall be concerned with both Fisher-KPP (monostable) nonlinearities and Allen-Cahn (bistable) nonlinearities. 1.Fisher-KPP: Reaction diffusion equations with logistic nonlinearity were introduced in the pioneer works of Fisher or Kolmogorov, Petrovsky and Piskunov. They are widely used in the literature to model phenomena arising in population genetics, or in biological invasions. The main property of such equations is to admit (biologically relevant) travelling wave solutions with some semi-infinite interval of admissible wave speed. Is is known that, under some assumptions on the initial data, the singular limit of the rescaled Fisher-KPP equation is an interface moving by constant speed which turns out to be the minimal speed of monotone related one-dimensional travelling waves. In this part of the project, we would like to address the following issues: a. What happens if one introduces a delay effect? b. What happens if one introduces a non-local effect? c. What is the real link between the convergence of the problem and the stability of the associated travelling waves? 2. Allen-Cahn: Allen and Cahn have introduced a microscopic diffusional theory for the motion of a curved antiphase boundary. The interfacial velocity is found to be linearly proportional to the mean curvature of the boundary and the constant of proportionality does not include the specific surface free energy. This new theory is confirmed by experimental measurements of domain coarsening kinetics in Fe---Al alloys. For even not well-prepared initial data, precise studies of both the generation and the motion of interface have shown that the rescaled Allen-Cahn equation converges to motion by mean curvature (MMC in short). In the classical MMC framework (i.e. for small times), fine estimates of the thickness of the transition layers are also known. In the viscosity framework, the convergence for all times to the generalized MMC defined by Evans-Spruck and Chen-Giga-Goto is proved. As far as Allen-Cahn equations are concerned, we would like to address the following issues. a. Can we estimate the thickness of the transition layers past the onset of singularities? b. What happens if we consider density-dependent diffusion? c. What happens if we replace the classical Laplace operator with the fractional Laplace operator? d. What happens when the comparison principle does not hold?

The aim of this project is to develop and strengthen collaboration among participants, to share complementary knowledge, and to pursue international collaborations in the transversal and competitive field of the project. Four of the six participants have got a new position in 2006 and one in 2005: the participants would like to expand this dynamic through this ANR project. The six participants are conducting their research on the deformation theory of algebraic or geometric structures appearing in representation theory or in mathematical physics. This is a vast area, touching upon many different subjects besides representation theory, such as topology, algebraic geometry, combinatorics, conformal field theory,... Recent years have seen spectacular results emerge from interaction between these fields, such as Witten's approach to the geometric Langlands program via four dimensional gauge theories, or Khovanov's construction of new quantum knot invariants using a mixture of topological quantum field theory and singularity theory. The participants all specialize in one or several aspects of the theory, and the knowledge the participants possess in the field will be illustrated by the detailed description of the different topics in which the participants intend to do research. The aim of this ANR project is to strengthen the collaboration that has emerged and is emerging among the participants; to give them the opportunity to share their different perspectives and to use these different points of views to obtain significantly new results; to organize international conferences of common interest; and to increase the international visibility of the French activity of young researchers in this field. All participants have worked or are working with internationally recognized experts of the field. The participants have already met several times to discuss this ANR project. A first research meeting (small CIRM research group) is already planned in Luminy in spring 2008.