Aggregation-diffusion equations: dynamics, asymptotics, and singular limits

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method -- the blob method for diffusion -- to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits

Spikes and diffusion waves in one-dimensional model of chemotaxis

We consider the one-dimensional initial value problem for the viscous transport equation with nonlocal velocity $u_t = u_{xx} - \left(u (K^\prime \ast u)\right)_{x}$ with a given kernel $K'\in L^1(\R)$. We show the existence of global-in-time nonnegative solutions and we study their large time asymptotics. Depending on $K'$, we obtain either linear diffusion waves ({\it i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion waves (the fundamental solution of the viscous Burgers equation) in asymptotic expansions of solutions as $t\to\infty$. Moreover, for certain aggregation kernels, we show a concentration of solution on an initial time interval, which resemble a phenomenon of the spike creation, typical in chemotaxis models

The AU Microscopii Debris Disk: Multiwavelength Imaging and Modeling

(abridged) Debris disks around main sequence stars are produced by the erosion and evaporation of unseen parent bodies. AU Microscopii (GJ 803) is a compelling object to study in the context of disk evolution across different spectral types, as it is an M dwarf whose near edge-on disk may be directly compared to that of its A5V sibling beta Pic. We resolve the disk from 8-60 AU in the near-IR JHK' bands at high resolution with the Keck II telescope and adaptive optics, and develop a novel data reduction technique for the removal of the stellar point spread function. The point source detection sensitivity in the disk midplane is more than a magnitude less sensitive than regions away from the disk for some radii. We measure a blue color across the near-IR bands, and confirm the presence of substructure in the inner disk. Some of the structural features exhibit wavelength-dependent positions. The disk architecture and characteristics of grain composition are inferred through modeling. We approach the modeling of the dust distribution in a manner that complements previous work. Using a Monte Carlo radiative transfer code, we compare a relatively simple model of the distribution of porous grains to a broad data set, simultaneously fitting to midplane surface brightness profiles and the spectral energy distribution. Our model confirms that the large-scale architecture of the disk is consistent with detailed models of steady-state grain dynamics. Here, a belt of parent bodies from 35-40 AU is responsible for producing dust that is then swept outward by the stellar wind and radiation pressures. We infer the presence of very small grains in the outer region, down to sizes of ~0.05 micron. These sizes are consistent with stellar mass-loss rates Mdot_* << 10^2 Mdot_sun.Comment: ApJ accepted, 56 pages, preprint style. Version in emulateapj with high-resolution figures available at http://tinyurl.com/y6ent

Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid

Using group theoretical methods, we analyze the generalization of a one-dimensional sixth-order thin film equation which arises in considering the motion of a thin film of viscous fluid driven by an overlying elastic plate. The most general Lie group classification of point symmetries, its Lie algebra, and the equivalence group are obtained. Similar reductions are performed and invariant solutions are constructed. It is found that some similarity solutions are of great physical interest such as sink and source solutions, travelling-wave solutions, waiting-time solutions, and blow-up solutions.Comment: 8 page

Asymptotic dynamics of attractive-repulsive swarms

We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel's first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady-state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly-supported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers' equation. We derive an analytical upper bound for the finite blow-up time after which the solution forms one or more $\delta$-functions.Comment: 23 pages, 10 figures; revised version updates the analysis in sec. 2.1 and 2.2, and contains enhanced discussion of the admissible class of social interaction force