The role of self-similarity in singularities of PDE's

We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a \textit{dynamical system}. We point out that analysing the dynamics close to the fixed point is a useful way of characterising the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. For each "class" of singularity, we give detailed examples.Comment: to be published in Nonlinearit

Optimal control for a two-sidedly degenerate aggregation equation

In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the wellposedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo–Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and first-order necessary conditions. The main novelty of this work concerns the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations

Biological Aggregation Driven By Social and Environmental Factors: A Nonlocal Model and Its Degenerate Cahn-Hilliard Approximation

Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. Beginning with a particular nonlocal aggregation model [Topaz et al., Bull. Math. Bio., 2006], we derive the minimal well-posed long-wave approximation, which is a degenerate Cahn-Hilliard equation. Energy minimizers of this reduced, local model retain many salient features of those of the nonlocal model, especially for large populations and away from an aggregation\u27s boundaries. Using the Cahn-Hilliard model as a testbed, we investigate the degree to which an external potential modeling food sources can be used to suppress peak population density, which is essential for controlling locust outbreaks. A random distribution of food sources tends to increase peak density above its intrinsic value, while a periodic pattern of food sources can decrease it

Recent Advances in Single-Particle Tracking: Experiment and Analysis

This Special Issue of Entropy, titled “Recent Advances in Single-Particle Tracking: Experiment and Analysis”, contains a collection of 13 papers concerning different aspects of single-particle tracking, a popular experimental technique that has deeply penetrated molecular biology and statistical and chemical physics. Presenting original research, yet written in an accessible style, this collection will be useful for both newcomers to the field and more experienced researchers looking for some reference. Several papers are written by authorities in the field, and the topics cover aspects of experimental setups, analytical methods of tracking data analysis, a machine learning approach to data and, finally, some more general issues related to diffusion