Simone Fagioli

Measure solutions for non-local interaction PDEs with two species

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.

A nonlocal swarm model for predators–prey interactions

We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator–prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.

Deterministic particle approximation of scalar conservation laws

In this paper we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. The result is complemented with some numerical simulations.

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

Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend the analysis to a system for two species interacting with each other according to different inner- and intra-species attractions. Under suitable conditions on this self- and crosswise attraction an interesting effect can be observed, namely phase separation into neighbouring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is simply connected. Preliminary results indicate the existence of phase separation, i.e. spatial sorting of the different species. A detailed analysis in one spatial dimension follows. The existence and shape of segregated stationary solutions is shown via the Krein-Rutman theorem. Moreover, for small repulsion/nonlinear diffusion, also uniqueness of these stationary states is proved.

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

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 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.

Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows

We review the recent results and present new ones on a deterministic follow-the-leader particle approximation of first-and second-order models for traffic flow and pedestrian movements. We start by constructing the particle scheme for the first-order Lighthill–Whitham–Richards (LWR) model for traffic flow. The approximation is performed by a set of ODEs following the position of discretized vehicles seen as moving particles. The convergence of the scheme in the many particle limit toward the unique entropy solution of the LWR equation is proven in the case of the Cauchy problem on the real line. We then extend our approach to the initial–boundary value problem (IBVP) with time-varying Dirichlet data on a bounded interval. In this case, we prove that our scheme is convergent strongly in \(\mathbf {L^{1}}\) up to a subsequence. We then review extensions of this approach to the Hughes model for pedestrian movements and to the second-order Aw–Rascle–Zhang (ARZ) model for vehicular traffic. Finally, we complement our results with numerical simulations. In particular, the simulations performed on the IBVP and the ARZ model suggest the consistency of the corresponding schemes, which is easy to prove rigorously in some simple cases.

A Deterministic Particle Approximation for Non-linear Conservation Laws

We review our analytical and numerical results obtained on the microscopic Follow-The-Leader (FTL) many particle approximation of one-dimensional conservation laws. More precisely, we introduce deterministic particle schemes for the Hughes model for pedestrian movements and for two vehicular traffic models that are the scalar Lighthill–Whitham–Richards model (LWR) and the \(2\times 2\) system Aw–Rascle–Zhang model (ARZ). Their approximation is performed by a set of ODEs, determining the motion of platoons of possible fractional vehicles or pedestrians seen as particles. Convergence results of the schemes in the many particle limit are stated. The numerical simulations suggest the consistency of the schemes.

Suitable weak solutions of the Navier–Stokes equations constructed by a space–time numerical discretization

We prove that weak solutions obtained as limits of certain numerical space-time discretizations are suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, in the space-periodic setting, we consider a full discretization in which the theta-method is used to discretize the time variable, while in the space variables we consider appropriate families of finite elements. The main result is the validity of the so-called local energy inequality.