Category Archives: Past seminar

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

 

Existence and stability of steady states of Flat Vlasov Poisson with a central mass density

Speaker: Matias Moreno

University of Chile

Date: Tuesday, November 15, 2022 at 12 Santiago time

Abstract: El Sistema de Vlasov-Poisson Plano es usado para modelar objetos astronómicos extremadamente planos, a través de la evolución de una distribución de partículas en el plano de fase, las cuales autointeractúan a través del campo gravitacional inducido por ellas mismas.

En este trabajo se estudia la construcción de estados estacionarios del Sistema de Vlasov-Poisson Plano en presencia de un potencial gravitacional externo, inducido por una densidad de masa fija. La construcción de dichos estados se realiza a través de la minimización de los funcionales de Energía-Casimir, los cuales permiten probar resultados de estabilidad no lineal para dichos estados estacionarios.

 

Venue: Online via Zoom / Sala multimedia CMM, 6th floor, Beauchef 851
Chair: Paola Rioseco

 

YouTube video (in Spanish)

 

Simulations of black hole binaries and their gravitational waves

Speaker: Tomas Andrade 

Instituto de Ciencias del Cosmos, Universidad de Barcelona

Date: Tuesday, November 22, 2022 at 12 Santiago time

Abstract: Our recently acquired ability to detect gravitational waves has expanded our senses and our possibilities of inquiring about the Universe. As a new era of gravitational wave detections rapidly unfolds, the importance of having accurate models for their signals becomes increasingly important. In this context, we will discuss numerical simulations of black hole binaries. In particular, we will focus on how they are made, what they help us achieve, and what are the current challenges in this research area.

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

Recent developments in Fourier interpolation theory

Speaker: Mateus Sousa

Basque Center for Applied Mathematics, Spain

Date: Tuesday, November 8, 2022 at 12 Santiago time

Abstract: In this talk we will discuss some problems related to the notion of Fourier interpolation:  formulas where one can recover functions from its values over a certain discrete set, and the values of its Fourier transform over a dual discrete set. These formulas arrive naturally in many situations, and we will mention a few related to certain kinds of uncertainty principles, and the theory of sphere packing.

This talk is meant for a broad audience with basic knowledge in analysis.

 

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

Chair: Rayssa Caju

Qualitative Properties for a class of fully nonlinear elliptic equations

Speaker: David Stolnicki

La Sapienza University of Rome, Italy

Date: Tuesday, October 27, 2022 at 2 p.m. Santiago time

Abstract: In joint work with F. Pacella, we study the existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci’s extremal operators in dimension two and higher. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover, for the (negative) Pucci P^- operator, we show the existence of a critical exponent and give bounds for it. The same technique is then applied in higher dimensions to improve the previously known bounds.

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

 

 

Existence and multiplicity of positive solutions to the scalar-field equation on large annuli in the three-dimensional sphere

Speaker: Satoshi Tanaka

Tohoku University, Japan

Date: Tuesday, October 25, 2022 at 12 Santiago time

Abstract: In this talk, we consider the Dirichlet problem for the scalar-field equation in a large annulus in the three-dimensional sphere. We obtain the existence, uniqueness, and multiplicity results of the positive solutions depending only on the latitude. This is joint work with Noaki Shioji (Yokohama National University) and Kohtaro Watanabe (National Defense Academy).

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Hanne Van Den Bosch

 

YouTube video (in English)

 

Inverse scattering for critical semilinear wave equations

Speaker: Gunther Uhlmann

University of Washington

Date: Tuesday, October 6, 2022 at 2 p.m. Santiago time

Abstract: In inverse scattering ione attempts to find the properties of a medium
by making remote observations. It has applications in physics,
geophysics, medical imaging, non-destructive evaluation of materials.
Radar and sonar are examples of inverse scattering methods that are
used routinely nowadays. In this case we consider the inverse problem
of determining the nonlinearity for  critical semilinear wave
equations. This is joint work with A, Sa Barreto and Y. Wang.

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

 

Multiplicidad de soluciones por cambios de magnitud

Speaker: Pilar Herreros

Pontificia Universidad Católica de Chile

Date: Tuesday, October 4, 2022 at 12 Santiago time

Abstract: Estudiaremos las soluciones radialmente simétricas del problema $$ \Delta u+f(u)=0,\quad x\in \mathbb{R}^N, N> 2,   \lim_{|x|\to \infty} u(x)=0. $$ Veremos que podemos generar nuevas soluciones del problema si introducimos cambios bruscos en la magnitud de la función f. Usando esto construiremos funciones f, definidas por partes, tales que el problema tiene cualquier número pre-determinado de soluciones.

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

YouTube video (in Spanish)