Finite energy scattering for the Lorentz-Maxwell equation

In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz-Maxwell equation for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation

The second iterate for the Navier-Stokes equation

We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak strong uniqueness for Koch-Tataru solutions; and finally an instability result in $B^{-1}_{\infty,q}$, for q>2.Comment: 13 page

Space-time resonances

This article is a short exposition of the space-time resonances method. It was introduced by Masmoudi, Shatah, and the author, in order to understand global existence for nonlinear dispersive equations, set in the whole space, and with small data. The idea is to combine the classical concept of resonances, with the feature of dispersive equations: wave packets propagate at a group velocity which depends on their frequency localization. The analytical method which follows from this idea turns out to be a very general tool.Comment: 10 page

Global existence for coupled Klein-Gordon equations with different speeds

Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.Comment: 35 page

Bilinear dispersive estimates via space-time resonances, part II: dimensions 2 and 3

Consider a bilinear interaction between two linear dispersive waves with a generic resonant structure (roughly speaking, space and time resonant sets intersect transversally). We derive an asymptotic equivalent of the solution for data in the Schwartz class, and bilinear dispersive estimates for data in weighted Lebesgue spaces. An application to water waves with infinite depth, gravity and surface tension is also presented.Comment: 45 page

Bilinear oscillatory integrals and boundedness for new bilinear multipliers

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger ; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.Comment: 35 pages, 3 figure

The KK-theory of amalgamated free products

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen.Comment: V.3, the paper has been splitted into two papers, this is the first part on amalgamated free product

THE DEMAND FOR INFORMAL FINANCE IN THE CONTEXT OF URBAN AFRICAN MARKETS: THE IMPORTANCE OF SOCIAL RELATIONS.

This paper provides an empirical analysis of the demand for financial services by micro-entrepreneurs operating in urban African markets. It particularly seeks to identify the motivations behind the use of the so-called informal financial practices in an environment where formal financial markets are active. The evidence provided in this paper is in sharp contradiction with the idea that that participation to informal finance is solely triggered by the lack of access to formal financial market. Using a first-hand dataset collected in three markets located in Ouagadougou, the capital city of Burkina Faso (West Africa), this paper shows that the use of some financial devices is, to some extent, dependent on motivations that are not financial in nature such as the need to create or enhance social relations.