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We concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound from below the blow-up rate for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than ${(T-t)}^{-1}$, the expected one. Moreover, we state that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.

In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than ${(T-t)}^{-1}$, the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.

Ce texte présente les résultats obtenus dans [BI11, BI14] en collaboration avec Liviu Ignat sur la représentation et les propriétés de dispersion de la solution de l’équation linéaire de Schrödinger sur certains graphes métriques. Le cas de l’équation de Schrödinger sur la droite avec plusieurs potentiels de Dirac découle comme cas particulier.

In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in ${ℝ}^3$ and it is used as a model for the evolution of a vortex filament in fluid mechanics. The main theorem give, under suitable assumptions, the existence and description of solutions generated by curves with a corner, for positive and negative times. Its companion theorem describes the evolution of perturbations...

On étudie la stabilité de la dynamique singulière de vortex filamentaire décrite dans [], qui engendre un coin en temps fini. On montre que sous certaines perturbations petites et régulières, le coin est encore formé. Notre approche utilise le flot binormal et la transformation de Hasimoto. On se ramène aux propriétés de scattering longue portée pour une équation de type Gross-Pitaesvski avec coefficients variables en temps. Ce travail a été obtenu en collaboration avec Luis Vega.

We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions ${\chi }_a(t,x)$ form a family of evolving regular curves in ${ℝ}^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We consider curves that are small regular perturbations of ${\chi }_a(t_0,x)$ for a fixed time $t_0$. In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...