Regularity and Long Time Behavior of Metric and Topological Models of Collective Dynamics

In the countryside of Northern Europe, there is stillness; a flock of starlings stand. Seemingly all at once, thousands of the small birds leap into the air. They fly in groups called murmurations, creating mesmerizing swirling shapes in the air. Each one makes complex aerial maneuvers dangerously close to their neighboring birds, yet never colliding; they continue their dance until they all come to rest once again. Deep below the surface of the ocean, a massive school of jack fish slowly swirl in the shape of a tornado, tightly knit so that the eye loses track of the individual. At each layer of the tornado, the fish swim in an alternating orientation, in the phenomenon known as milling. Deep within a minuscule embryo, cells replicate, as their number grows they aggregate along the edge of the tiny sphere that is the beginnings of what will be a new creature. Atop looming skyscrapers, a small group of pigeons are perched. A wind blows, the pigeons flow with the wind, together they fly to their next rooftop. These occurrences all have a common factor; the emergence of global phenomena from local inter- actions. These are just a few of the many spectacles that collective behavior attempts to model. The mathematics of collective behavior is a relatively new and emerging area of mathematics. Since the 1990s various models of collective motion have come about. Among these are the Vicsek model that can describe the sort of milling models seen in the behavior of schools of fish; the Kuramoto synchronization model, which can represent the way neurons in the brain fire, or how heart cells beat; and the Cucker-Smale model that exhibits the flocking and alignment behavior often observed in birds. There are three levels at which the mathematics of collective behavior is performed. The first is the discrete or microscopic level, where you have a finite number of agents, like birds or fish. As you let the number of agents grow, individuals can become indistinguishable, and the second level of the model arises; the kinetic formulation or mesoscopic level, which can describe bacteria. The third and final level is found by taking the moments of the kinetic formulation. Through integrating we get the hydrodynamic or macroscopic level. At each of these levels the mathematics of collective behavior is concerned with proving well-posedness of the specific models of interest. By this we mean the existence, uniqueness, and continuous dependence on initial data of solutions. Further, the models should exhibit the properties that are observed in the systems they are meant to represent. In general this means proving the diameter of the group of agents remains finite (flocking) and that agents in a flock end up going in the same direction and at the same speed (alignment). In this thesis we focus on two of the above levels. First, the discrete level, where we look at the Cucker-Smale system appended with an external force. The force here is representative of friction and self-propulsion, where the friction keeps agents from moving too fast, and the self-propulsion serves as the ability of an agent to move itself. The inclusion of this external force, though more physically realistic, leads to the loss of important properties, like conservation of momentum and Galilean invariance. Without these properties, analysis becomes much more difficult. In Chapter 1 the method of Grassmannian reduction is introduced in order to prove alignment of velocity for a certain class of solutions to the model. In Chapter 2 the method of Grassmanian reduction is used to derive an application to opinion dynamics in a game theoretic framework. In this chapter the existence and uniqueness of a Nash equilibrium is proved. Further the equilibrium is proved to be a global attractor for the system showing the convergence of opinions to a best ’agreement’ possible, or a compromise. In Chapter 3 the notion of topological distance and adaptive diffusion is introduced at the discrete level. Globabl well-posedness of this Cucker-Smale system is shown and conditional alignment, with connectivity conditions, is given similar to the classical theory. Formal derivation of the kinetic and hydrodynamic topological models is shown. In Chapter 4 we look at the topologicial Euler alignment model, the hydrodynamic analog of the Cucker-Smale system discussed in Chapter 3, and prove local well-posedness of this model in higher order Sobolev spaces using coercivity estimates, and by looking at the viscous regularization of the system and applying a fixed point argument. In Chapter 5 we look at the topological Euler alignment system from Chapter 4, but appended with the friction force from Chapter 1. For the special case of p = 1, we prove alignment of unidirectional flows.