Important: Please note that there has been a change of room for the talks of Friday, March 31st. During the week of 27-31 March 2023 we will hold a course and a few talks by visiting researchers on the topic of nonlinear and nonlocal partial differential equations motivated by some problems in biology and collective behaviour. The course will be given by José Antonio Carrillo on Monday 27, Tuesday 28 and Thursday 30, suited for graduate students working in analysis, PDE or mathematical modelling. The program of the talks can be found below.
Schedule
Monday, March 27 (Seminar 2, IMAG):
11:00 - 13:30 - Course by Carrillo
Tuesday, March 28 (Seminar 2, IMAG):
11:00 - 13:30 - Course by Carrillo
Thursday, March 30 (Seminar 2, IMAG):
11:30 - 14:00 - Course by Carrillo
15:30 - 16:15 - Alejandro Fernández
16:15 - 17:00 - Mauricio Sepúlveda
Friday, March 31 (Aula A24, Facultad de Ciencias):
(Within "Seminario de Ecuaciones Diferenciales"). These talks can also be followed online: https://ugr.zoom.us/my/mastermat2, password 9L73vK
12:00 - 12:45 - José Antonio Carrillo
12:45 - 13:30 - David Gómez Castro
Scientific program
Speaker: José Antonio Carrillo (Mathematical Institute, University of Oxford)
Title: Short course: Nonlocal Aggregation-Diffusion Equations
Abstract: I will give an overview of the state of the art of global minimizers of free energies with two homogeneous competing effects: nonlinear diffusion modelling repulsion and nonlocal terms modelling attraction. I will discuss the results on the corresponding gradient flows associated with these free energies leading to aggregation-diffusion equations. I will particularly focus on understanding their long-time asymptotics and qualitative properties.
Speaker: Alejandro Fernández (University of Oxford)
Title:A gradient flow approach to study a family of Cahn-Hilliard equations
Abstract: We will discuss about the family of Cahn-Hilliard equations
$$
\partial_t = - div (\rho \nabla (\Delta \rho)) - \Delta \rho^m,
$$
where \(1 \leq m \leq 2 + \frac{2}{d}\) and \(d\) is the dimension. This family of equations has applications in biology, for two species it models cell-cell adhesion. Furthermore, they also occur in connection with many physical systems involving fluid interfaces, for example, it describe fluid droplets hanging from a ceiling. Our goal is to prove existence of very weak solutions for our family of Cahn-Hilliard equations. In order to do that, we take advantage of a gradient flow approach based on a time-discretization scheme with the 2-Wasserstein distance. This method was developed by Jordan, Kinderlehrer and Otto (JKO scheme) in the late 90s to study the Fokker-Planck equation. In this talk we will introduced this method, and we will use it for our case.
The talk presents on-going work with J.A. Carrillo, A. Esposito and C. Falcó.
Speaker: Mauricio Sepúlveda (Universidad de Concepción)
Title: Inverse Problem for an intestinal crypt model
Abstract: We consider an intestinal crypt model including microbiota-derived regulations. The simplified model considers a coupled system of 2 degenerate parabolic equations with cross diffusion whose unknowns are the density of progenitor cells (pc) and stem cells (sc). Additionally, the density of deep crypt secretory (DCS) cells acts as a function that we can assume to be known and that is known to affect the population dynamics in the crypt. The inverse problem consists in determining the parameters that define the shape of the density function of the DCS cells (slopes and position), from partial measurements of stem and progenitor cells. For this, we propose a classical method of adjoint state.
The general intestinal crypt model (considering 4 cell types) was introduced by Beatrice Laroche from INRAE, France, and her PhD student Marie Haghebaert who has used BGK schemes to successfully simulate the dynamics of the phenomenon.
Speaker: José Antonio Carrillo (Mathematical Institute, University of Oxford)
Title: Nonlocal Aggregation-Diffusion Equations: fast diffusion and partial concentration
Abstract: We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.
Speaker: David Gómez Castro (Universidad Complutense de Madrid)
Title: Viscosity solutions for aggregation-diffusion problems
Abstract: The theory of viscosity solutions was developed to deal with Hamilton-Jacobi type problems in the late 80s and 90s by Crandall, Lions, and others. The surname "viscosity" comes from their construction through the vanishing viscosity method. In many scenearios have well-posedness and select the physical solution in many settings. Furthermore, their stability properties makes them suitable to study different approximations: finite differences and asymptotic limits as $t \to \infty$.
The aim of this talk is to introduce this notion of solution and show its usefulness to study the time limit of Aggregation-Diffusion equations.
The talk presents joint work with: J.A. Carrillo, A. Fernández-Jiménez, and J.L. Vázquez
List of participants
(Speakers not listed here)
- Prasanta Barik (Universidad de Granada)
- Inmaculada Benítez (Universidad de Granada)
- María J. Cáceres (Universidad de Granada)
- José Cañizo (Universidad de Granada)
- Samuel Charles (University of Oxford)
- Shuchen Guo (University of Oxford)
- Basim Mustafa (Universidad de Granada)
- Francisco Javier Reyes (Universidad de Granada)
- Niccolò Tassi (Universidad de Granada)
- Nicolás Torres (Universidad de Granada)
