Global Existence and Stability of Nearly Aligned Flocks
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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.
KeywordsFlocking Alignment Cucker–Smale Fractional Laplacian
Mathematics Subject Classification92D25 35Q35 76N10
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
- 9.Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing III: fractional diffusion of order \(0<\alpha <1\). Phys. D (accepted)Google Scholar