The seminar takes place on Fridays, at 14:15 in SR 2.040.
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Title: Effective dynamics and Blow Up in a model for Magnetic Relaxation
Since the works of L. Woltjer (1958) and J. B. Taylor (1974-1986), it is believed in the physical community that a low resistivity magneto fluid (like that of the corona of the Sun) relaxes due to viscosity to a static, force-free state. Mathematically speaking, the so-called Taylor conjecture is far from being understood, due to the complexity of the Magnetohydrostatic equations (MHD) used to model this scenario. This is why different effective models have been proposed to study this phenomena, in an attempt to understand whether the Taylor conjecture holds and, if so, how such relaxed state is attained. In this talk, we present the mathematical analysis of a simple model for Magnetic Relaxation proposed by H.K. Moffat in 2015. The low resistivity assumption is modeled via a singular perturbation problem, in which we prove the existence of a two-stage evolution, with a fast time scale followed by a slower second stage. Furthermore, we provide an effective equation for the second one and show that such system can undergo a blow-up in finite time, opening the door to new phenomena not foreseen when this model was proposed. This is joint work with Dimitri Cobb and Juan J. L. Velázquez.
Schedule
April 18
No seminar
April 25
Hendrik Schröders
May 02
No Seminar
May 09
Javier Durán Fernández
May 16
Elena Demattè
May 23
No Seminar
May 30
No seminar
June 06
Dr. Eugenia Franco
June 13
TBA
June 20
TBA
June 27
No seminar
July 4
TBA
July 11
Max Mihailescu
July 18
Daniel Sánchez-Simón del Pino (joint session)
July 25
TBA
Past talks
Speaker: Jann Hendrik Schröders
Title: Mass transport in Fokker-Planck equations
We consider Fokker-Planck equations in several dimensions with general nonconvex potentials that have multiple local minima. In the one-dimensional case, the vanishing diffusion limit is governed by the so called "Kramers' law", which yields simple and expicit reaction-diffusion dynamics. In the talk, we extend this to the higher dimensional setting and describe a graph structure for the limit system. There, the limit dynamics are much more sensitive to the geometry of the potential, which can change the topology of the underlying graph.
Speaker: Javier Durán Fernández
Title: A Fermionic Model Related to Random Forests
The arboreal gas measure is the uniform measure on spanning forests of weighted graphs. We discuss the percolative properties of the arboreal gas and its phase transition in the infinite-volume limit. Due to its combinatorial nature, this measure is difficult to study directly. Instead, it is more convenient to consider a related anticommuting (fermionic) model. We introduce Grassmann variables to define this model, which can then be analyzed using tools from mathematical physics.
Speaker: Elena Demattè
Title: A free boundary problem involving radiation
In this talk we consider a free boundary problem for the melting of ice where we assume that the heat is transported by both conduction and radiation. Specifically, we study a one-dimensional, two-phase, Stefan-like problem which contains a non-local integral operator. After briefly discussing well-posedness results, we will show the existence of traveling wave solutions, which is not the case in the classical Stefan problem. The properties of these solutions will be studied using maximum-principle methods, blow-up limits and Liouville-type Theorems for non-linear equations. This is joint work with J. L. L. Velázquez.
Speaker: Dr. Eugenia Franco
Title: Non equilibrium chemical networks and their biological functions
The detailed balance is a property of macroscopic systems that are obtained from an underlying time reversible microscopic model. It states that each elementary process (for instance, each chemical reaction) is in equilibrium with its reverse process. Even if, at the fundamental level, we expect chemical reactions to satisfy the so-called detailed balance condition, biochemical systems are often modeled by systems of equations for which detailed balance fails. This can be justified if the system is in contact with "reservoirs" that are out of equilibrium, as it is usually the case in biological systems. In the first part of the talk, I will show that every chemical reaction network that is endowed with mass action kinetics and that does not satisfy the detailed balance property can be obtained freezing the concentration of some substances in a chemical network for which the detailed balance property hold. I will then discuss the relation between the detailed balance property and the adaptation property. The adaptation property, which is present in many chemical networks, states that the network reacts to a stimulus, but returns to the pre-stimulus state after a transient time. This work has been done in collaboration with Prof. Juan J. L. Velázquez.
Speaker: Max Mihailescu
Title: Non equilibrium chemical networks and their biological functions
In this talk, I will introduce the Spin-1 Ising model (sometimes called Blume Capel), which can be thought of as an Ising model with an additional "zero" spin in between +1 and -1. A mean-field computation suggests that the zero phase should dominate at any positive temperature, which is in contrast to the (regular) Ising model, where a phase transition is known to occur in all dimensions greater than 2. I will sketch the proof for the absence of symmetry breaking in the two dimensional model, which uses a percolation type argument. If time permits, I will further explain why the proof fails in dimension 3 and higher. This is joint work with Margherita Disertori, Ron Peled y Tom Spencer.
