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Full-Text Articles in Physical Sciences and Mathematics
Asymptotic Dynamics Of Attractive-Repulsive Swarms, Andrew J. Leverentz '08, Chad M. Topaz, Andrew J. Bernoff
Asymptotic Dynamics Of Attractive-Repulsive Swarms, Andrew J. Leverentz '08, Chad M. Topaz, Andrew J. Bernoff
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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. …
Blowup And Dissipation In A Critical-Case Unstable Thin Film Equation, Thomas P. Witelski, Andrew J. Bernoff, Andrea L. Bertozzi
Blowup And Dissipation In A Critical-Case Unstable Thin Film Equation, Thomas P. Witelski, Andrew J. Bernoff, Andrea L. Bertozzi
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We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.