Organizers: Giovanni Catino and Fabio Cipriani
Nicolas Zadeh, Université Libre de Bruxelles, Belgium,
Description and numerical study of a kinetic Fokker-Planck equation in neuroscience, Monday, July 01, 2024, time 15:15, Aula seminari MOX, VI piano
Abstract:Abstract:
The well-known Integrate and fire model in a partial differential equation form has been at the center of many mathematical developments since Brunel's work and the seminal paper by Caceres, Carrillo and Perthame in 2011. Its descriptive shortcomings such as the inability to see sub-threshold oscillations or to obtain resonances gave birth to the resonate and fire model (Izhikevich, 2001), phenomenologically complex enough but computationally not costly.
However, there has been no deep mathematical study of it yet. In this work, we first establish a PDE corresponding to the mean-field limit of a population of resonate and fire neurons.
The obtained formulation corresponds to a non-linear kinetic Fokker-Planck equation, with a non-local linearity and a measure source term, studied on a half plane. Even though the obtained operator has properties of hypoellipticity, the theoretical study is tedious, encouraging the pursuit of a numerical study to obtain information about the behaviour of the solutions.
We thereby will describe the positivity and mass preserving finite differences scheme of experimental order one we developed, which allows us to observe all the properties we were expecting from the original single neuron model, and even giving birth to some conjectures which we shall detail.
Francesca Anceschi, Università Politecnica delle Marche,
Well-posedness results for Kolmogorov equations with applications to mean-field control problems for multi-agent systems, Thursday, May 16, 2024, time 15:00, Aula Seminari - III Piano
Abstract:Abstract:
In this talk, based on a joint project with G. Ascione, D. Castorina and F. Solombrino, we discuss some well-posedness results for Kolmogorov-Fokker-Planck equations with measurable coefficients in time and locally Holder continuous coefficients in space with possibly unbounded drift terms and their application to the study of particle systems of the second order whose prototypical agent is driven by a McKean-Vlasov SDE, or by a Vlasov-Fokker-Planck PDE.
Eduardo Munoz Hernandez, Universidad Complutense de Madrid,
Existence and multiplicity of coexistence states in a heterogeneous predator-prey model with saturation, Tuesday, May 14, 2024, time 11:30, Aula Seminari - III Piano
Abstract:Abstract:
In this talk we analyze a predator-prey model coming from the heterogeneous counterparts of the classical Lotka-Volterra
and Holling-Tanner models by considering a non-negative saturation term that can vanish inside the domain. By the use of the maximum principle and bifurcation theory the regions of coexistence can be exactly determined. Furthermore, by varying the amplitude of the saturation different multiplicity results will be obtained. This is a joint work with Julián López-Gómez.
Alessio Falocchi, Politecnico di Milano,
On the long-time behaviour of solutions to unforced evolution Navier-Stokes equations under Navier boundary conditions, Thursday, May 09, 2024, time 16:30, Aula Seminari - III Piano
Abstract:Abstract:
We study the asymptotic behaviour of the solutions to Navier-Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest that we call sectors. The main motivations come from the celebrated results by Foias-Saut related to the long time behaviour of the solutions to Navier-Stokes equations under Dirichlet conditions.
Here the choice of the boundary conditions requires carefully considering the geometry of the domain, due to the possible lack of the Poincaré inequality in presence of axial symmetries. In non-axially symmetric domains we show the validity of the Foias-Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias-Saut result holds for initial data belonging to one of the invariants.
This is a joint work with Prof. Elvise Berchio (Politecnico di Torino, Italy) and Clara Patriarca (Université libre de Bruxelles, Belgium).
Marco Caroccia, Politecnico di Milano,
On the singular planar plateau problem, Tuesday, May 07, 2024, time 15:00, Aula Seminari - III Piano
Abstract:Abstract:
The classical Plateau problem asks which surface in three-dimensional space spans the least area among all the surfaces with boundary given by an assigned curve S. This problem has many variants and generalizations, along with (partial) answers, and has inspired numerous new ideas and techniques. In this talk, we will briefly introduce the problem in both its classical and modern contexts, and then we will focus on a specific vectorial (planar) type of the Plateau problem. Given a curve S in the plane, we can ask which diffeomorphism T of the disk D maps the boundary of D to S and spans the least area, computed as the integral of the Jacobian of T, among competitors with the same boundary condition. For simply connected curves, the answer is provided by the Riemann map, and the minimal area achieved is the Lebesgue measure of the region enclosed by S. For more complex curves, possibly self-intersecting, new analysis is required. I will present a recent result in this sense, obtained in collaboration with Prof. Riccardo Scala from the University of Siena, where the value of the minimum area is computed with an explicit formula that depends on the topology of S.
Elena Danesi, Università di Padova,
Strichartz estimates for the Dirac equation on compact manifolds without boundary, Thursday, April 18, 2024, time 15:00, Aula seminari - III piano
Abstract:Abstract:
The Dirac equation on Rn can be listed within the class of dispersive equations, together with, e.g., the wave and Klein-Gordon equations. In the years a lot of tools have been developed in order to quantify the dispersion of a system. Among these one finds the Strichartz estimates, that are a priori estimates of the solutions in mixed Lebesgue spaces. For the flat case Rn they are known, as they are derived from the ones that hold for the wave and Klein-Gordon equations. However, when passing to a curved spacetime domain, very few results are present in the literature. In this talk I will firstly introduce the Dirac equation on curved domains. Then, I will discuss the validity of this kind of estimates in the case of Dirac equations on compact Riemannian manifolds without boundary. This is based on a joint work with Federico Cacciafesta (Università di Padova) and Long Meng (CERMICS-École des ponts ParisTech).