Large time behavior of nonlocal aggregation models with nonlinear diffusion
Martin Burger Marco Di Francesco
Networks & Heterogeneous Media 2008, 3(4): 749-785 doi: 10.3934/nhm.2008.3.749
The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, one obtains concentrated densities (Dirac $\delta$ distributions) as stationary solutions and large time limits in the absence of diffusion. In addition, we provide a comparison for aggregation kernels with infinite respectively finite support. In the first case, there is a unique stationary solution corresponding to concentration at the center of mass, and all solutions of the evolution problem converge to the stationary solution for large time. The speed of convergence in this case is just determined by the behavior of the aggregation kernels at zero, yielding either algebraic or exponential decay or even finite time extinction. For kernels with finite support, we show that an infinite number of stationary solutions exist, and solutions of the evolution problem converge only in a measure-valued sense to the set of stationary solutions, which we characterize in detail.
Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients.
All these results are obtained via a reformulation of the equations considered using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension.
keywords: Wasserstein metric asymptotic behavior biological aggregation nonlocal PDEs stationary solutions Nonlinear diffusion
Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models
Marco Di Francesco Donatella Donatelli
Discrete & Continuous Dynamical Systems - B 2010, 13(1): 79-100 doi: 10.3934/dcdsb.2010.13.79
In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller-Segel type systems. The approximating systems are either hyperbolic-parabolic or hyperbolic-elliptic. They all feature a nonlinear pressure term arising from a volume filling effect which takes into account the fact that cells do not interpenetrate. The main convergence result relies on energy methods and compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assumptions, we also prove an existence result for initial data which are small perturbation of a constant state. Such result is proven via classical Friedrichs's symmetrization and linearization. In order to simplify the coverage, we restrict to the two-dimensional case with periodical boundary conditions.
keywords: Keller Segel model chemotaxis diffusive relaxation Singular convergence volume filling effect. nonlinear diffusion nonlinear hyperbolic systems
Deterministic particle approximation of the Hughes model in one space dimension
Marco Di Francesco Simone Fagioli Massimiliano Daniele Rosini Giovanni Russo
Kinetic & Related Models 2017, 10(1): 215-237 doi: 10.3934/krm.2017009

In this paper we present a new approach to the solution to a generalized version of Hughes' models for pedestrian movements based on a follow-the-leader many particle approximation. In particular, we provide a rigorous global existence result under a smallness assumption on the initial data ensuring that the trace of the solution along the turning curve is zero for all positive times. We also focus briefly on the approximation procedure for symmetric data and Riemann type data. Two different numerical approaches are adopted for the simulation of the model, namely the proposed particle method and a Godunov type scheme. Several numerical tests are presented, which are in agreement with the theoretical prediction.

keywords: Crowd dynamics conservation laws eikonal equation Hughes' model for pedestrian flows particle approximation
Mean field games with nonlinear mobilities in pedestrian dynamics
Martin Burger Marco Di Francesco Peter A. Markowich Marie-Therese Wolfram
Discrete & Continuous Dynamical Systems - B 2014, 19(5): 1311-1333 doi: 10.3934/dcdsb.2014.19.1311
In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.
keywords: mean field limit optimal control numerical simulations. Pedestrian dynamics calculus of variations
Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion
Marco Di Francesco Yahya Jaafra
Kinetic & Related Models 2019, 12(2): 303-322 doi: 10.3934/krm.2019013

In this paper we consider a one-dimensional nonlocal interaction equation with quadratic porous-medium type diffusion in which the interaction kernels are attractive, nonnegative, and integrable on the real line. Earlier results in the literature have shown existence of nontrivial steady states if the $L^1$ norm of the kernel $G$ is larger than the diffusion constant $\varepsilon$. In this paper we aim at showing that this equation exhibits a 'multiple' behavior, in that solutions can either converge to the nontrivial steady states or decay to zero for large times. We prove the former situation holds in case the initial conditions are concentrated enough and 'close' to the steady state in the $∞$-Wasserstein distance. Moreover, we prove that solutions decay to zero for large times in the diffusion-dominated regime $\varepsilon≥ \|G\|_{L^1}$. Finally, we show two partial results suggesting that the large-time decay also holds in the complementary regime $\varepsilon < \|G\|_{L^1}$ for initial data with large enough second moment. We use numerical simulations both to validate our local asymptotic stability result and to support our conjecture on the large time decay.

keywords: Quadratic diffusion nonlocal interaction asymptotic behavior steady states Wasserstein distance
Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic
Marco Di Francesco Simone Fagioli Massimiliano D. Rosini
Mathematical Biosciences & Engineering 2017, 14(1): 127-141 doi: 10.3934/mbe.2017009

We consider the follow-the-leader approximation of the Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. We prove rigorous convergence to weak solutions of the ARZ system in the many particle limit in presence of vacuum. The result is based on uniform ${\mathbf{BV}}$ estimates on the discrete particle velocity. We complement our result with numerical simulations of the particle method compared with some exact solutions to the Riemann problem of the ARZ system.

keywords: Aw-Rascle-Zhang model second order models for vehicular traffics many particle limit
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior
Marco Di Francesco Alexander Lorz Peter A. Markowich
Discrete & Continuous Dynamical Systems - A 2010, 28(4): 1437-1453 doi: 10.3934/dcds.2010.28.1437
We study the system

$ c_t+u \cdot \nabla c = \Delta c- nf(c) $
$ n_t + u \cdot \nabla n = \Delta n^m- \nabla \cdot (n \chi(c)\nabla c) $
$ u_t + u \cdot \nabla u + \nabla P - \eta\Delta u + n \nabla \phi=0 $
$\nabla \cdot u = 0. $

arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous--medium--like diffusion in the equation for the density $n$ of the bacteria, motivated by a finite size effect. We prove that, under the constraint $m\in(3/2, 2]$ for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case $m=2$ we prove that solutions converge to constant states in the large--time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case $m=1$. The case $m=2$ is very special as we can provide a Lyapounov functional. We generalize our results to the three--dimensional case and obtain a smaller range of exponents $m\in$( m*$,2]$ with m*>3/2, due to the use of classical Sobolev inequalities.

keywords: nonlinear diffusion Stokes equations. chemotaxis model

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