We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞.
A (Patlak-) Keller-Segel model in two space dimensions with an additional crossdiffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross diffusion, the global existence of weak solutions to the parabolic-parabolic model as well as the global existence of bounded weak solutions to the parabolic-elliptic model, thus preventing blow up of the cell density. Furthermore, the long-time decay of the solutions to the parabolic-elliptic model is shown and finite-element simulations are presented illustrating the influence of the regularizing cross-diffusion term.
In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite direction. The pedestrian dynamics are driven by aversion and cohesion, i.e. the tendency to follow individuals from the own group and step aside in the case of contraflow. We start with a 2D lattice based approach, in which the transition rates reflect the described dynamics, and derive the corresponding PDE system by formally passing to the limit in the spatial and temporal discretization. We discuss the existence of special stationary solutions, which correspond to the formation of directional lanes and prove existence of global in time bounded weak solutions. The proof is based on an approximation argument and entropy inequalities. Furthermore we illustrate the behavior of the system with numerical simulations.
A reactive kinetic transport equation whose macroscopic limit is the KPP-Fisher equation is considered. In a scale, where collisions occur at a faster rate than reactions, existence of traveling waves close to those of the KPP-Fisher equation is shown. The method adapts a micro-macro decomposition in the spirit of the work of Caflisch and Nicolaenko for the Boltzmann equation. Stability of these waves is shown for perturbations in a weighted L 2-space, where the weight function is exponential and such that the (macroscopic) linearized operator in the weighted space is self-adjoint and negative definite. Similar approaches to stability of traveling waves are wellknown for the KPP-Fisher equation.
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