Category Archives: Past seminar

Fourier transform restriction phenomena and applications to control of dispersive equations

Past seminar

Speaker: Roberto Capistrano-Filho

Universidade Federal de Pernambuco – Brasil

Date: Tuesday, October 10th, 2023 at 12 Santiago time

Abstract: In 93′ Jean Bourgain discovered a subtle smoothing property of solutions of the KdV equation posed on a periodic domain. Since this celebrated article, the smoothing properties for dispersive systems are now well known. In this talk, we will see that in the last 15 years, the Bourgain spaces are fundamental to addressing the global control problems in periodic frameworks. Precisely, we will show that the propagation of compactness and regularity have been observed thanks to these spaces in various control problems for dispersive systems.

Venue:  Sala John Von Neumann, 7th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

Variational techniques for non-trivial boundary conditions in fluid dynamics and beyond

Past seminar

Speaker: Nicolás Barnafi

CMM – Universidad de Chile

Date: Monday, October 2nd, 2023 at 12 Santiago time

Abstract: Despite the widely spread usage of no-slip boundary conditions in fluid dynamics, there exist several applications where it is not true that the fluid sticks to the boundary. Instead, the natural condition is setting the normal flux to zero, and then complementing the tangential direction with certain friction conditions known as the Navier boundary condition. In this talk, we will explore the many issues that can arise from strongly imposed boundary conditions in this setting, and how such issues can be alleviated in a way that additionally allows for extreme scale numerical simulations.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

Travelling wave solutions to abcd systems

Past seminar

Speaker: Olivier Goubet

Université de Lille – CNRS, France

Date: Monday, September 11th, 2023 at 12 Santiago time

Abstract:  We are interested in
the so-called \( abcd\) system introduced by J. Bona, M. Chen and JC. Saut.
This system reads $$h_{t}+v_x-bh_{txx}+av_{xxx}+(hv)_x=0,$$
$$v_{t}+h_x-dv_{txx}+ch_{xxx}+vv_x=0.$$ Here  we revisit the existence and stability results for travelling waves solutions of these systems.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

Switching Controllabilty for Parabolic system

Past seminar

Speaker: Felipe Chaves-Silva

Universidade Federal da Paraíba

Date: Monday, September 4th, 2023 at 12 Santiago time

Abstract: In this talk, we discuss controllability properties of systems under switching structures. More precisely, we focus on the problem of controlling parabolic systems with several actuators under the constraint that at any given time at most one control is active on the system. We give necessary and sufficient condition, depending only on the underlying operators, for the swtiching controllability to hold.

Venue:  Sala de Seminarios, 5th floor, Beauchef 851 / Online via Zoom
Chair: Rayssa Caju

Algunas soluciones bi y tridimensionales de mapas armónicos inspirados en la biología y la astrofísica

Past seminar

Speaker: Axel Osses 

DIM- CMM, Universidad de Chile

Date: Monday, August 21st, 2023 at 12 Santiago time

Abstract: Presentamos algunas soluciones exactas y aproximadas de mapas armónicos débiles de las ecuaciones no lineales de Frank-Oseen provenientes de la teoría de cristales líquidos y sus posibles aplicaciones en el marco de la identificación no invasiva de la dirección de fibras musculares en el corazón humano, así como la sorprendente similitud que tienen con ciertas estructuras en en campo de la astrofísica.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

Reversal in the Stationary Prandtl Equations

Past seminar

Speaker: Sameer Iyer

University of California – Davis, USA.

Date: Monday, August 14th, 2023 at 12 Santiago time

Abstract: We investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which \(u > 0\) and \(u < 0\). The classical point of view of regarding the Prandtl equations as an evolution \(x\) completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

Energy decay for classes of nonlocal dispersive equations

Past seminar

Speaker: Ricardo Freire

DIM CMM – Universidad de Chile

Date: Monday, August 7th, 2023 at 12 Santiago time

Abstract: We consider the long-time dynamics of large solutions to a special class of evolution equations. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times. Moreover, in the case of interior regions, we prove decay for a sequence of times. The classes of nonlocal dispersive equations which we will treat are as follows:

$$\begin{cases} \partial_t u + L_\alpha u + u\partial_x u=0, \quad x,t\in \mathbb{R}, \\u(x,0)=u_0(x)\end{cases}$$

where \(\alpha>0\),and the operator \(L_\alpha\) is the Fourier multiplier operator by a real-valued odd function belonging to \((C^1(\mathbb{R})\cap C^\infty(\mathbb{R}^∗))\). These classes contain, in particular, the following equations: the fractional KdV, Benjamin-Ono and the Intermediate Long Wave, for example.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

Sobre las propiedades de continuación única discreta y sus aplicaciones al control y problemas inversos

Past seminar

Speaker: Jaime Ortega

DIM CMM – Universidad de Chile

Date: Monday, July 10, 2023 at 12 Santiago time

Abstract: Es bien conocido que las Propiedades de Continuación Única juegan un importante papel en el estudio de los problemas de controlabilidad y problemas inversos. También es natural preguntarse que sucede con las discretizaciones de las EDP y sus propiedades. En esta charla mostraremos que al considerar la discretización, en diferencias finitas, de operadores diferenciales, este tipo de propiedades pierden su validez, por lo que es válido preguntarse si resultados de controlabialidad, identificabilidad y estabilidad en los problemas inversos siguen siendo válidos en el caso discreto. Aquí damos una respuesta afirmativa a esta pregunta, pero la validez de estos resultados dependen de ciertos parámetros relacionados con el tamaño de la malla.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

New mathematical model for Tsunamis with precise time arrival predictions

Past seminar

Speaker: Khawla Msheik

Université Lyon 1 – Institut Camille Jordan, France

Date: Monday, July 3, 2023 at 12 Santiago time

Abstract: We propose in this work a new system of equations modeling Tsunamis. It is a coupled system accounting for both water compressibility and viscoelasticity of the earth. Adding these latter physical effects is responsible for the closest-to-reality time arrival predictions (among existing models), capturing the negative peak before the main wave hump, and the exhibition of the negative dispersion phenomena. This comes in remarkable agreement with previous experiments and studies on the topic. The system is also delivered in a relatively simple mathematical structure of equations that is easy to solve numerically. Further well posedness results are also investigated.

Venue: Online via Zoom / Sala de Seminarios, 5th floor, Beauchef 851
Chair: Rayssa Caju

On the collapse of the local Rayleigh condition for the hydrostatic Euler equations

Past seminar

Speaker: Victor Cañulef 

CMM – Universidad de Chile

Date: Monday, June 19, 2023 at 12 Santiago time

Abstract: The hydrostatic Euler equations are derived from the incompressible Euler equations by means of the hydrostatic approximation. Among the different stability criteria that arise in the study of linear stability for the incompressible Euler equations, we can mention Rayeligh’s stability criterion, which gives rise to the local Rayleigh condition. Linear and nonlinear instability of the hydrostatic Euler equations around certain shear flows is well-known, as well as the finite time blow-up of certain solutions that do not satisfy the local Rayleigh condition. On the other hand, local existence, uniqueness and stability has been established in Sobolev spaces under the local Rayleigh condition. In this talk I will present new features of the \(H^4\) solution to the hydrostatic Euler equations under the local Rayleigh condition; under certain assumptions, we establish the dichotomy between the breakdown of the local Rayleigh condition and the formation of singularities. Additionally, we get necessary conditions for global solvability in Sobolev spaces. As a byproduct, we show the $x-$independence of stationary solutions. Our proof relies on new monotonicity identities for the solution to the hydrostatic Euler equations under the local Rayleigh condition.

Venue: Online via Zoom / Sala John Von Neumann, 7th floor, Beauchef 851
Chair: Rayssa Caju