Category Archives: Past seminar

Uniform a priori estimates for positive solutions of the Lane-Emden equation and system in the plane

Speaker: Boyan Sirakov

PUC-Rio, Brazil

Date: Monday, March 6, 2023 at 12 Santiago time

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 is compelling evidence of the richer nature of the system case. We prove the partial result that uniform boundedness holds provided the exponents in the system are comparable (while many open questions subsist). As a consequence, the energy of the solutions is uniformly bounded, and this has similar consequences as for the scalar equation.

This is a joint work with N. Kamburov from PUC-Chile.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

Long time asymptotics of large data in the Kadomtsev-Petviashvili models and geometrical aspects of its dynamics

Speaker: Felipe Poblete

Universidad Austral de Chile, Chile

Date: Tuesday, January 10, 2023 at 12 Santiago time

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.

Joint work with A. J. Mendez, C. Muñoz and J.C. Pozo

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

On traveling waves for the Gross-Pitaevskii equations

Speaker: André de Laire

University of Lille, France

Date: Tuesday, January 3, 2023 at 12 Santiago time

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 by variational techniques. Then, we will discuss the existence and behavior of the dark solitons for the Gross-Pitaevskii equation is a strip, according to its width.
This is joint work with Philippe Gravejat, Salvador Lopez-Martinez, and Didier Smets.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

Neural Implicit Surface Evolution using Differential Equations

Speaker: Tiago Novello

IMPA, Brasil

Date: Tuesday, December 20, 2022 at 12 Santiago time

Abstract: In this talk, we present a machine learning framework that uses smooth neural networks to model dynamic variations of implicit surfaces under partial differential equations.

Examples include evolving an initial surface towards vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of implicit surfaces regularized by specific differential equations.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

Cosmología Primordial

Speaker: Gonzalo Palma 

Universidad de Chile

Date: Tuesday, December 20, 2022 at 12 Santiago time

Abstract: Nuestro entendimiento del origen del universo ha cambiado dramáticamente durante los últimos 40 años. Hoy sabemos que la estructura de gran escala del universo (compuesto por galaxias) debe su existencia a pequeñas fluctuaciones del espacio y el tiempo -fluctuaciones primordiales- ya presentes durante el Big-Bang. La teoría más aceptada para explicar el origen de estas fluctuaciones sostiene que ellas se deben a procesos cuánticos ocurridos antes del Big-Bang, durante una época conocida como inflación cósmica. Este cuadro lo hemos forjado utilizando observaciones basadas en mensajeros electromagnéticos (luz), pero gracias a la reciente detección de ondas gravitacionales estamos transitando hacia una nueva era de la cosmología también basada en mensajeros gravitacionales. En esta charla voy a introducir los conceptos básicos que utilizamos para investigar nuestro universo primordial, y describiré avances recientes (experimentales y teóricos) para entender el origen de las fluctuaciones primordiales.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Claudio Muñoz

Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

Speaker: Delia Schiera 

Instituto Superior Técnico, Lisbon

Date: Thursday, December 6, 2022 at 12 Santiago time

Abstract:  I will present some recent results obtained in collaboration with A. Pistoia and H. Tavares for a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and critical nonlinearities. We show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem, and we establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. I will also briefly present existence of least-energy solutions for the particular case in which the system reduces to a biharmonic equation, and some symmetry results in the case the domain is an annulus.

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Gabrielle Nornberg

 

YouTube video (in English)

 

Non-existence results for an eigenvalue problem involving Lipschitzian nonlinearities with non-positive primitive and applications

Speaker: Olivier Goubet 

University of Lille, France

Date: Thursday, December 1, 2022 at 14:15 Santiago time

Abstract: We discuss here existence and non existence results for nonlinear eigenvalues problems like

$$\Delta u = \lambda \sin u,  \lambda\geq 0$$

in a bounded domain of \( \mathbb{R}^D\) with homogeneous Dirichlet conditions.
We then infer some applications for the long time behavior of solutions to sine-Gordon equations. This work is a joint work with B. Ricceri (Catane, Italy).

Venue: Online via Zoom / Sala seminarios DIM, 5th floor, Beauchef 851
Chair: Claudio Muñoz

 

 

Nonlocal Aggregation-Difusion Equations: entropies, gradient flows, phase transitions and applications

Speaker: José Carrillo

University of Oxford, United Kingdom

Date: Tuesday, November 29, 2022 at 12 Santiago time

Abstract: This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and nancial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear difusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear di usion, one can work in the L2 framework, nonlinear difusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient ow structure of these problems. The theoretical analysis of the asymptotic stability of the diferent branches of solutions is a challenging open problem.

This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang.

This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

On Space-Time Formulations and Boundary Integral Equations for the Wave Equation

Speaker: Carolina Urzúa

Delft University of Technology

Date: Tuesday, November 24, 2022 at 12 Santiago time

Abstract: Space-time discretization methods are becoming increasingly popular, since they allow adaptivity in space and time simultaneously, and can use parallel iterative solution strategies for time-dependent problems. However, in order to exploit these advantages, one needs to have a complete numerical analysis of the corresponding Galerkin methods.

In this talk, we consider the wave equation with on a Lipschitz bounded domain, with either Dirichlet or Neumann boundary conditions, and with zero initial conditions. The first step to build the required numerical analysis was to show new existence and uniqueness results for the weak formulations of these initial boundary value problems. With this, we are able to propose a new approach to boundary integral equations for the wave equation that allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise stale and efficient adaptive space-time boundary element methods.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

From sign-changing solutions of the Yamabe equation to critical competitive systems

Speaker: María Medina

Universidad Autónoma de Madrid

Date: Tuesday, November 17, 2022 at 14:15 Santiago time

Abstract: In this talk we will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find positive solutions to critical competitive systems in dimension 4.

Venue: Online via Zoom / Sala de seminarios DIM, 5th floor, Beauchef 851
Chair: Claudio Muñoz

 

YouTube video (in Spanish)