arXiv subjects

Gianluca Favre

Publications and source records attributed to Gianluca Favre.

8 recordsLinked to original sources

arXiv2026

A framework for continuum modeling of opinion dynamics on a network based on probability of connections

We propose a modeling framework to develop a continuum description of opinion dynamics on networks as an alternative to other models, like the ones based on graphons. In a nutshell, the continuum model that we propose aims at approximating the distribution of opinions as well as the probability that two given opinions are connected. To illustrate our framework, we focus on a simple model of consensus dynamics on a network and derive a continuum description using techniques inspired by mean-field limits. We also discuss the limitations of this approach and suggest extensions to account for dynamic networks with evolving connections, stochastic effects, and directional interactions.

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arXiv2024

Stability analysis for a kinetic bacterial chemotaxis model

We perform stability analysis of a kinetic bacterial chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous tumbling kernel represents the key challenge for the stability analysis as it rules out a direct linearization of the nonlinear terms. To address this challenge we fruitfully separate the evolution of the shape of the cellular profile from its global motion. We provide a full nonlinear stability theorem in a perturbative setting when chemical degradation can be neglected. With chemical degradation we prove stability of the linearized operator. In both cases we obtain exponential relaxation to equilibrium with an explicit rate using hypocoercivity techniques. To apply a hypocoercivity approach in this setting, we develop two novel and specific approaches: i) the use of the $H^1$ norm instead of the $L^2$ norm, and ii) the treatment of nonlinear terms. This work represents an important step forward in bacterial chemotaxis modeling from a kinetic perspective as most results are currently only available for the macroscopic descriptions, which are usually parabolic in nature. Significant difficulty arises due to the lack of regularization of the kinetic transport operator as compared to the parabolic operator in the macroscopic scaling limit.

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arXiv2023

Opinion formation on evolving network. The DPA method applied to a nonlocal cross-diffusion PDE-ODE system

We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the 'attitude areas,' which depend on the distance between the agents' opinions. The position of the agents on the network evolves based on the distance between the agents' opinions. The goal is to study radicalization, polarization, and fragmentation of the population while changing its open-mindedness and the radius of interaction.

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arXiv2022

Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model

A reaction-kinetic model for a two-species gas mixture undergoing pair generation and recombination reactions is considered on a flat torus. For dominant scattering with a non-moving constant-temperature background the macroscopic limit to a reaction-diffusion system is carried out. Exponential decay to equilibrium is proven for the kinetic model by hypocoercivity estimates. This seems to be the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model as well as the first hypocoercivity result for a nonlinear kinetic problem without smallness assumptions. The analysis profits from uniform bounds of the solution in terms of the equilibrium velocity distribution.

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arXiv2021

Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit

The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law. For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.

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arXiv2020

Existence analysis of a degenerate diffusion system for heat-conducting fluids

The existence of global weak solutions to a parabolic energy-transport system in a bounded domain with no-flux boundary conditions is proved. The model can be derived in the diffusion limit from a kinetic equation with a linear collision operator involving a non-isothermal Maxwellian. The evolution of the local temperature is governed by a heat equation with a source term that depends on the energy of the distribution function. The limiting model consists of cross-diffusion equations with an entropy structure. The main difficulty is the nonstandard degeneracy, i.e., ellipticity is lost when the fluid density or temperature vanishes. The existence proof is based on a priori estimates coming from the entropy inequality and the $H^{-1}$ method and on techniques from mathematical fluid dynamics (renormalized formulation, div-curl lemma).

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arXiv2020

Hypocoercivity and Fast Reaction Limit for Linear Reaction Networks with Kinetic Transport

The long time behavior of a model for a first order, weakly reversible chemical reaction network is considered, where the movement of the reacting species is described by kinetic transport. The reactions are triggered by collisions with a nonmoving background with constant temperature, determining the post-reactional equilibrium velocity distributions. Species with different particle masses are considered, with a strong separation between two groups of light and heavy particles. As an approximation, the heavy species are modeled as nonmoving. Under the assumption of at least one moving species, long time convergence is proven by hypocoercivity methods for the cases of positions in a flat torus and in whole space. In the former case the result is exponential convergence to a spatially constant equilibrium, and in the latter it is algebraic decay to zero, at the same rate as solutions of parabolic equations. This is no surprise since it is also shown that the macroscopic (or reaction dominated) behavior is governed by the diffusion equation.

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arXiv2019

On a Navier-Stokes-Allen-Cahn model with inertial effects

A mathematical model describing the flow of two-phase fluids in a bounded container $Ω$ is considered under the assumption that the phase transition process is influenced by inertial effects. The model couples a variant of the Navier-Stokes system for the velocity $u$ with an Allen-Cahn-type equation for the order parameter $φ$ relaxed in time in order to introduce inertia. The resulting model is characterized by second-order material derivatives which constitute the main difficulty in the mathematical analysis. Actually, in order to obtain a tractable problem, a viscous relaxation term is included in the phase equation. The mathematical results consist in existence of weak solutions in 3D and, under additional assumptions, existence and uniqueness of strong solutions in 2D. A partial characterization of the long-time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed.

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