Marco Di Francesco

Large time behavior of nonlocal aggregation models with nonlinear diffusion

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

The Keller–Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion

\delta

Nonlinear Cross-Diffusion with Size Exclusion

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.

Measure solutions for non-local interaction PDEs with two species

The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller–Segel model of chemotaxis with volume filling effect. In the linear diffusion case we provide several sufficient conditions for the diffusion part to dominate and yield decay to zero solutions. We also provide an explicit decay rate towards self–similarity. Moreover, we prove that no stationary solutions with positive mass exist. In the nonlinear diffusion case we prove that the asymptotic behavior is fully determined by whether the diffusivity constant in the model is larger or smaller than the threshold value

The entropy dissipation method for spatially inhomogeneous reaction-diffusion-type systems

\varepsilon =1

A nonlocal swarm model for predators–prey interactions

. Below this value we have existence of nondecaying solutions and their convergence (along subsequences) to stationary solutions. For

Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling Species

\varepsilon >1

Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic.

all compactly supported solutions are proved to decay asymptotically to zero, unlike in the classical models with linear diffusion, where the asymptotic behavior depends on the initial mass.

Optimal L1 rate of decay to diffusion waves for the Hamer model of radiating gases

The aim of this paper is to investigate the mathematical properties of a continuum model for diffusion of multiple species incorporating size exclusion effects. The system for two species leads to nonlinear cross-diffusion terms with double degeneracy, which creates significant novel challenges in the analysis of the system. We prove global existence of weak solutions and well-posedness of strong solutions close to equilibrium. We further study some asymptotics of the model, and in particular we characterize the large-time behavior of solutions.

On a mean field game optimal control approach modeling fast exit scenarios in human crowds

This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems of non-local interaction PDEs with two species, which are the PDE counterpart of systems of deterministic interacting particles with two species. The main motivations behind those models arise in cell biology, pedestrian movements, and opinion formation. In case of symmetrizable systems (i.e. with cross-interaction potentials one multiple of the other), we provide a complete existence and uniqueness theory within (a suitable generalization of) the Wasserstein gradient flow theory in Ambrosio et al (2008 Gradient Flows in Metric Spaces and in the Space of Probability Measures (Lectures in Mathematics ETH Zurich) 2nd edn (Basel: Birkhauser)) and Carrillo et al (2011 Duke Math. J. 156 229–71), which allows the consideration of interaction potentials with a discontinuous gradient at the origin. In the general case of non-symmetrizable systems, we provide an existence result for measure solutions which uses a semi-implicit version of the Jordan–Kinderlehrer–Otto (JKO) scheme (Jordan et al 1998 SIAM J. Math. Anal. 29 1–17), which holds in a reasonable non-smooth setting for the interaction potentials. Uniqueness in the non-symmetrizable case is proven for C2 potentials using a variant of the method of characteristics.