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- W3136285805 abstract "We overview recent developments in the study of alignment hydrodynamics, driven by a general class of symmetric communication kernels. A main question of interest is to characterize the emergent behavior of such systems, which we quantify in terms of the spectral gap of a weighted Laplacian associated with the alignment operator. Our spectral analysis of energy fluctuation covers both long-range and short-range kernels and does not require thermal equilibrium (no closure for the pressure). In particular, in the prototypical case of metric-based short-range kernels, the spectral gap admits a lower-bound expressed in terms of the discrete Fourier coefficients of the radial kernel, which enables us to quantify an emerging flocking behavior for non-vacuous solutions. These large-time behavior results apply as long as the solutions remain smooth. It is known that global smooth solutions exist in one and two spatial dimensions, subject to sub-critical initial data. We settle the question for arbitrary dimension, obtaining non-trivial initial threshold conditions which guarantee existence of multiD global smooth solutions." @default.
- W3136285805 created "2021-03-29" @default.
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- W3136285805 date "2021-04-01" @default.
- W3136285805 modified "2023-10-14" @default.
- W3136285805 title "On the Mathematics of Swarming: Emergent Behavior in Alignment Dynamics" @default.
- W3136285805 doi "https://doi.org/10.1090/noti2254" @default.
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