Noticias en español
Boundary controllability of critically singular parabolic equations on convex domains.
Abstract: In this talk I will discuss the null boundary control of heat-like equations on convex domains, featuring a singular potential that diverges as the inverse square of the distance to the boundary. For this purpose, I will establish global Carleman estimates for the associated operators by combining intermediate inequalities with distinct weights that involve non-smooth powers of the boundary distance. These estimates are sharp in the sense that they capture both the natural boundary conditions and the -energy for the problem. Additionally, I will describe the role of the potential...
Read MoreDynamics of Concentrated Vorticities In 2d and 3d Euler Flows.
Abstract: A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions’ existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG. In particular we establish Helmholtz’ conjecture on...
Read MoreUniform a priori estimates for positive solutions of the Lane-Emden equation and system in the plane.
Abstract A few years ago we proved that positive solutions of the superlinear Lane-Emden equation in a two-dimensional smooth bounded domain are bounded independently of the exponent in the equation. Apart from being interesting in itself, this information plays a pivotal role in the asymptotic study of solutions for large exponents, as well as contributes to the old and hard conjecture of uniqueness of positive solutions in a convex domain. We recently took up a similar study for the Lane-Emden system and discovered that, contrary to initial intuition, the boundedness fails in general. This...
Read MoreDirichlet-to-Neumann and Calderon operator via deep learning techniques.
Abstract: In this talk we consider the Dirichlet-to-Neumann operator and the Calderón mapping appearing in Calderon’s inverse problem. Using deep learning techniques, we prove that these maps are rigorously approximated by infinite-dimensional neural networks.
Read MoreLong time asymptotics of large data in the Kadomtsev-Petviashvili models and geometrical aspects of its dynamics.
Abstract: In this talk we consider the Kadomtsev-Petviashvili equations posed on R2. For both models, we provide sequential in time asymptotic descriptions of solutions obtained from arbitrarily large initial data, inside and far regions of the plane not containing lumps or line solitons, and under minimal regularity assumptions. A geometrical description of the dynamics will be given in terms of parabolic regions.
Read MoreOn traveling waves for the Gross-Pitaevskii equations.
Abstract: In this talk, we will discuss some properties of traveling waves solutions for some variants of the classical Gross-Pitaevskii equation in the whole space, in order to include new physical models in Bose-Einstein condensates and nonlinear optics. We are interested in the existence of finite energy localized traveling waves solutions with nonvanishing conditions at infinity, i.e. dark solitons. After a review of the state of the art in the classical case, we will show some results for a family of Gross-Pitaevskii equations with nonlocal interactions in the potential energy, obtained...
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