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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References
Carrillo, J.A., Choi, Y.-P., Tadmor, E., Tan, C.: Critical thresholds in 1D Euler equations with non-local forces. Math. Models Methods Appl. Sci. 26(1), 185–206 (2016)
Constantin, P., Zelati, M.C., Vicol, V.: Uniformly attracting limit sets for the critically dissipative SQG equation. Nonlinearity 29(2), 298–318 (2016)
Constantin, P., Vicol, V.: Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22(5), 1289–1321 (2012)
Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)
Do, T., Kiselev, A., Ryzhik, L., Tan, C.: Global regularity for the fractional Euler alignment system. Arch. Ration. Mech. Anal. 228(1), 1–37 (2018)
Ha, S.-Y., Kang, M.-J., Kwon, B.: A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid. Math. Models Methods Appl. Sci. 24(11), 2311–2359 (2014)
He, S., Tadmor, E.: Global regularity of two-dimensional flocking hydrodynamics. C. R. Math. Acad. Sci. Paris 355(7), 795–805 (2017)
Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)
Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing III: fractional diffusion of order \(0<\alpha <1\). Phys. D (accepted)
Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing. Trans. Math. Appl. 1(1), tnx001 (2017)
Shvydkoy, R., Tadmor, E.: Eulerian dynamics with a commutator forcing II: flocking. Discrete Contin. Dyn. Syst. 37(11), 5503–5520 (2017)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)
Vicsek, T., Zefeiris, A.: Collective motion. Phys. Rep. 517, 71–140 (2012)
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