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.

The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the Keller-Segel model of chemotaxis with volume filling effect. In both cases we first cover the global existence and uniqueness theory of solutions of the Cauchy problem on R d. Then, we address the large time asymptotic behavior. In the linear diffusion case we provide several sufficient conditions such that the diffusion part dominates and yields decay to zero of 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 behaviour is fully determined by the diffusivity constant in the model being larger or smaller than the threshold value ε = 1. Below this value we have existence of non-decaying solutions and their convergence (in terms of subsequences) to stationary solutions. For ε > 1 all compactly supported solutions are proven to decay asymptotically to zero, unlike in the classical models with linear diffusion, where the asymptotic behaviour depends on the initial mass.

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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 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.

Abstract. 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. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in L 1 loc ) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in L ∞ , nonnegative, and with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, e.g. in the Lighthill-Whitham-Richards model for traffic flow) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect L ∞ → BV for nonlinear scalar conservation laws is intrinsic of the discrete model. Keywords: Micro-macro limit and Scalar conservation laws and Follow-the-leader models and Oleinik condition and Entropy solutions and Particle method. 35L65 and 35L45 and 90B20 and 65N75 and 82C22 . AMS Subject classification:

We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diffusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient flow generated by the energy E, which is the sum of a quadratic free energy and the interaction energy. The interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diffusivity constant is strictly smaller than the total mass of the interaction kernel. In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the Krein-Rutman theorem. The steady states are symmetric with respect to their center of mass x0, compactly supported on sets of the form [x0 − L, x0 + L], C 2 on their support, strictly decreasing on (x0, x0 +L). Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations.

In this paper we investigate the mathematical theory of Hughes' model for the flow of pedestrians (cf. Hughes (2002) [17]), consisting of a non-linear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task. For such an approximated system we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kružkov (1970) [22], in which the boundary conditions are posed following the approach of Bardos et al. (1979) [7]. We use BV estimates on the density ρ and stability estimates on the potential φ in order to prove uniqueness. Furthermore, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behavior of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.

We study the systemarising 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 ∈ (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 ∈ (m * , 2] with m * > 3/2, due to the use of classical Sobolev inequalities.