Sabine Hittmeir

Cross Diffusion Preventing Blow-Up in the Two-Dimensional Keller–Segel Model

A (Patlak–)Keller–Segel model in two space dimensions with an additional cross-diffusion 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.

CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL

A parabolic-parabolic (Patlak-) Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term 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 concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show

Traveling Waves of a Kinetic Transport Model for the KPP-Fisher Equation

L^\infty

Lane Formation by Side-Stepping

bounds for the solutions to the parabolic-elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results.

GLOBAL WELL-POSEDNESS FOR PASSIVELY TRANSPORTED NONLINEAR MOISTURE DYNAMICS WITH PHASE CHANGES

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

Derivation and analysis of continuum models for crossing pedestrian traffic

L^2

Asymptotics for moist deep convection I: Refined scalings and self-sustaining updrafts

-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 well known for the KPP-Fisher equation.

A conservative reconstruction scheme for the interpolation of extensive quantities in the Lagrangian particle dispersion model FLEXPART

In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite directions. The pedestrian dynamics are driven by aversion and cohesion, i.e., the tendency to follow individuals from their own group and step aside in the case of contraflow. We start with a two-dimensional 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.

Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics

We study a moisture model for warm clouds that has been used by Klein and Majda as a basis for multiscale asymptotic expansions for deep convective phenomena. These moisture balance equations correspond to a bulk microphysics closure in the spirit of Kessler and of Grabowski and Smolarkiewicz, in which water is present in the gaseous state as water vapor and in the liquid phase as cloud water and rain water. It thereby contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. Phase changes are associated with enormous amounts of latent heat and therefore provide a strong coupling to the thermodynamic equation. In this work we assume the velocity field to be given and prove rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture model with viscosity, diffusion and heat conduction. To guarantee local well-posedness we first need to establish local existence results for linear parabolic equations, subject to the Robin boundary conditions on the cylindric type of domains under consideration. We then derive a priori estimates, for proving the maximum principle, using the Stampacchia method, as well as the iterative method by Alikakos to obtain uniform boundedness. The evaporation term is of power law type, with an exponent in general less or equal to one and therefore making the proof of uniqueness more challenging. However, these difficulties can be circumvented by introducing new unknowns, which satisfy the required cancellation and monotonicity properties in the source terms.

On nonlinear conservation laws with a nonlocal diffusion term

In this paper, we study hyperbolic and parabolic nonlinear partial differential equation models, which describe the evolution of two intersecting pedestrian flows. We assume that individuals avoid collisions by sidestepping, which is encoded in the transition rates of the microscopic 2D model. We formally derive the corresponding mean-field models and prove existence of global weak solutions for the parabolic model. Moreover we discuss stability of stationary states for the corresponding one-dimensional model. Furthermore we illustrate the rich dynamics of both systems with numerical simulations.