## Dispersive blow-up and persistence properties for the Schrödinger-Korteweg-de Vries system.

Abstract: In this talk, we shall prove the existence of dispersive blow-up for the Schrödinger-Korteweg-de Vries system. Roughly, dispersive blow-up has being called to the development of point singularities due to the focussing of short or long waves. In mathematical terms, we show that the existence of this kind of singularities is provided by the linear dispersive solution by proving that the Duhamel term is smoother. This result is the first one regarding systems of nonlinear dispersive equations. To obtain our results we use, in addition to smoothing properties, persistence properties...

Read More## Connecting orbits in Hilbert spaces and applications to P.D.E.

Abstract: Functional Analysis methods are often useful to solve efficiently P.D.E. problems. The idea is to view a solution of a P.D.E. as a map with values in a space of functions, and reduce the initial P.D.E. to an O.D.E. problem. I will establish a general theorem on the existence of heteroclinic orbits in Hilbert spaces. As a first application, I will present a new construction, in a more general setting, of the heteroclinic double layers (initially constructed by Schatzman). As a second application, I will show the existence of the heteroclinic double layers for a fourth order...

Read More## Profile of touch-down solution to a non-local MEMS model

Abstract: In this talk, I am at showing the construction of solutions with extinction (touch-down) for the equation of MEMS devices (Micro-electromechanical systems). In addition to that, we also describe the asymptotic behavior of the solution at the extinction domain.

Read More## Existence of infinitely many solutions for the Einstein-Lichnerowicz system

Abstract: “We will consider in this talk the Einstein-Lichnerowicz system of equations. It originates in General Relativity as a way to determine initial-data sets for the evolution problem. This system takes the form of a strongly coupled, supercritical, nonlinear system of elliptic PDEs. We will investigate its blow-up properties and show that, under some assumptions on the physics data, it possesses a non-compact family of solutions. This family of solutions will be constructed by combining toplogical methods with a finite-dimensional reduction approach; due to the non-variational...

Read More## The variation of maximal functions on Euclidean spaces and spheres.

Abstract: Maximal operators are central objects of study in harmonic analysis. The most classical example is the Hardy-Littlewood maximal operator (in both its centered and uncentered form) but there are also other interesting ones, for instance the ones associated to a convolution with a smooth kernel. In this introductory talk we will introduce the sub-area of harmonic analysis that studies how the variation of a maximal function behaves with respect to the initial data. We will discuss the classical results in this topic (most of them dealing with the one dimensional case) and...

Read More## Unique continuation for some nonlinear dispersive models

ABSTRACT : This talk is concerned with unique continuation properties (UCP) for solutions to some time evolution equations. We shall study two types of UCP (1) local and (2) asymptotic at infinity. Roughly, (1) local means : If u, v are solutions of the equation which agree in an open set D, then they are identical in the whole domain of definition. Roughly, (2) asymptotic at infinity means if u, v are solutions such that ||| u(t)-v(t)|||<\Infty for t=t_1,and t=t_2, then they are identical in the whole domain of definition. The class of dispersive model to be considered includes the...

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