Abstract
We study regularity of a hydrodynamic singular model of collective behavior introduced in Shvydkoy and Tadmor (Trans Math Appl 1(1):tnx001, 2017). In this note we address the question of global well-posedness in multi-dimensional settings. It is shown that any initial data \((u,\rho )\) with small velocity variations \(|u(x) - u(y)| < \varepsilon \) relative to its higher order norms, gives rise to a unique global regular solution which aligns and flocks exponentially fast. Moreover, we prove that the limiting flocks are stable.
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Acknowledgements
Research was supported in part by NSF grant DMS 1515705, and the College of LAS at UIC. The author thanks Eitan Tadmor for stimulating discussions
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Shvydkoy, R. Global Existence and Stability of Nearly Aligned Flocks. J Dyn Diff Equat 31, 2165–2175 (2019). https://doi.org/10.1007/s10884-018-9693-8
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DOI: https://doi.org/10.1007/s10884-018-9693-8