arXiv subjects

Gaspard Jankowiak

Publications and source records attributed to Gaspard Jankowiak.

14 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.

math.AP↗

arXiv2024

Conservation, convergence, and computation for evolving heterogeneous elastic wires

The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.

math.AP↗

arXiv2022

The influence of nucleus mechanics in modelling adhesion-independent cell migration in structured and confined environments

Recent biological experiments have shown that certain types of cells are able to move in structured and confined environment even without the activation of focal adhesion. Focusing on this particular phenomenon and based on previous works, we derive a novel two-dimensional mechanical model, which relies on the following physical ingredients: the asymmetrical renewal of the actin cortex supporting the membrane, resulting in a backward flow of material; the mechanical description of the nucleus membrane and the inner nuclear material; the microtubule network guiding nucleus location; the contact interactions between the cell and the external environment. The resulting fourth order system of partial differential equations is then solved numerically to conduct a study of the qualitative effects of the model parameters, mainly those governing the mechanical properties of the nucleus and the geometry of the confining structure. Coherently with biological observations, we find that cells characterized by a stiff nucleus are unable to migrate in channels that can be crossed by cells with a softer nucleus. Regarding the geometry, cell velocity and ability to migrate are influenced by the width of the channel and the wavelength of the external structure. Even though still preliminary, these results can be potentially useful in determining the physical limit of cell migration in confined environment and in designing scaffold for tissue engineering.

math.AP↗

arXiv2022

Canards in a bottleneck

In this paper we investigate the stationary profiles of a nonlinear Fokker-Planck equation with small diffusion and nonlinear in- and outflow boundary conditions. We consider corridors with a bottleneck whose width has a global nondegenerate minimum in the interior. In the small diffusion limit the profiles are obtained constructively by using methods from geometric singular perturbation theory (GSPT). We identify three main types of profiles corresponding to: (i) high density in the domain and a boundary layer at the entrance, (ii) low density in the domain and a boundary layer at the exit, and (iii) transitions from high density to low density inside the bottleneck with boundary layers at the entrance and exit. Interestingly, solutions of the last type involve canard solutions generated at the narrowest point of the bottleneck. We obtain a detailed bifurcation diagram of these solutions in terms of the in- and outflow rates. The analytic results based on GSPT are further corroborated by computational experiments investigating corridors with bottlenecks of variable width.

math.DS↗

arXiv2021

Bifurcation of elastic curves with modulated stiffness

We investigate the equilibrium configurations of closed planar elastic curves of fixed length, whose stiffness, also known as the bending rigidity, depends on an additional density variable. The underlying variational model relies on the minimization of a bending energy with respect to shape and density and can be considered as a one-dimensional analogue of the Canham-Helfrich model for heterogeneous biological membranes. We present a generalized Euler-Bernoulli elastica functional featuring a density-dependent stiffness coefficient. In order to treat the inherent nonconvexity of the problem we introduce an additional length scale in the model by means of a density gradient term. We derive the system of Euler-Lagrange equations and study the bifurcation structure of solutions with respect to the model parameters. Both analytical and numerical results are presented.

math.AP↗

arXiv2020

A PDE model for unidirectional flows: stationary profiles and asymptotic behaviour

In this paper, we investigate the stationary profiles of a convection-diffusion model for unidirectional pedestrian flows in domains with a single entrance and exit. The inflow and outflow conditions at both the entrance and exit as well as the shape of the domain have a strong influence on the structure of stationary profiles, in particular on the formation of boundary layers. We are able to relate the location and shape of these layers to the inflow and outflow conditions as well as the shape of the domain using geometric singular perturbation theory. Furthermore, we confirm and exemplify our analytical results by means of computational experiments.

math.AP↗

arXiv2020

Micro- and Macroscopic Modeling of Crowding and Pushing in Corridors

Experiments with pedestrians revealed that the geometry of the domain, as well as the incentive of pedestrians to reach a target as fast as possible have a strong influence on the overall dynamics. In this paper, we propose and validate different mathematical models at the micro- and macroscopic levels to study the influence of both effects. We calibrate the models with experimental data and compare the results at the micro -- as well as macroscopic levels. Our numerical simulations reproduce qualitative experimental features on both levels, and indicate how geometry and motivation level influence the observed pedestrian density. Furthermore, we discuss the dynamics of solutions for different modeling approaches and comment on the analysis of the respective equations.

math.AP↗

arXiv2020

Modelling adhesion-independent cell migration

A two-dimensional mathematical model for cells migrating without adhesion capabilities is presented and analyzed. Cells are represented by their cortex, which is modelled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modelled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.

math.AP↗

arXiv2018

Non-Conforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part II: error estimates for periodic microstructure

This paper is dedicated to the rigorous numerical analysis of a Multiscale Finite Element Method (MsFEM) for the Stokes system, when dealing with highly heterogeneous media, as proposed in [B.P.~Muljadi et al., arXiv:1404.2837]. The method is in the vein of the classical Crouzeix-Raviart approach. It is generalized here to arbitrary sets of weighting functions used to enforced continuity across the mesh edges. We provide error bounds for a particular set of weighting functions in a periodic setting, using an accurate estimate of the homogenization error. Numerical experiments demonstrate an improved accuracy of the present variant with respect to that of Part I, both in the periodic case and in a broader setting.

math.NA↗

arXiv2016

Onofri inequalities and rigidity results

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

math.AP↗

arXiv2015

The Moser-Trudinger-Onofri inequality

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or Onofri inequality for brevity. In dimension two this inequality plays a role similar to the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, we state several elementary remarks. We also prove various new results. We give a proof of the inequality using mass transportation methods (in the radial case), consistently with similar results for Sobolev's inequalities. We investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, we establish that the inequality is an entropy--entropy production inequality, which provides an integral remainder term. Finally we give a proof of the inequality based on rigidity methods and introduce a related nonlinear flow.

math.AP↗

arXiv2014

Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities

This work focuses on an improved fractional Sobolev inequality with a remainder term involving the Hardy-Littlewood-Sobolev inequality which has been proved recently. By extending a recent result on the standard Laplacian to the fractional case, we offer a new, simpler proof and provide new estimates on the best constant involved. Using endpoint differentiation, we also obtain an improved version of a Moser-Trudinger-Onofri type inequality on the sphere. As an immediate consequence, we derive an improved version of the Onofri inequality on the Euclidean space using the stereographic projection.

math.FA↗

arXiv2014

Sobolev and Hardy-Littlewood-Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy-Littlewood-Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin-Talenti functions.

math.AP↗

arXiv2013

Stationary solutions of Keller-Segel type crowd motion and herding models: multiplicity and dynamical stability

In this paper we study two models for crowd motion and herding. Each of the models is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated to such solutions. The dynamical stability in a neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter and all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss qualitative properties of the solutions using theoretical methods and numerical computations.

math.AP↗