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Eulerian dynamics with a commutator forcing Ⅱ: Flocking

  • * Corresponding author: Eitan Tadmor

    * Corresponding author: Eitan Tadmor
Research was supported in part by NSF grant DMS 1515705 (RS) and by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR grant N00014-1512094 (ET)
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  • We continue our study of one-dimensional class of Euler equations, introduced in [11], driven by a forcing with a commutator structure of the form $[{\mathcal L}_φ, u](ρ)$, where $u$ is the velocity field and ${\mathcal L}_φ$ belongs to a rather general class of convolution operators depending on interaction kernels $φ$.

    In this paper we quantify the large-time behavior of such systems in terms of fast flocking, for two prototypical sub-classes of kernels: bounded positive $φ$'s, and singular $φ(r) = r^{-(1+α)}$ of order $α∈ [1, 2)$ associated with the action of the fractional Laplacian ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(·, t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.

    Mathematics Subject Classification: Primary: 35Q35, 76N10; Secondary: 92D25.


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  •   J.A. Carrillo , Y.-P. Choi , E. Tadmor  and  C. Tan , Critical thresholds in 1D Euler equations with non-local forces, Math. Models Methods Appl. Sci., 26 (2016) , 185-206.  doi: 10.1142/S0218202516500068.
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      T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155.
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      R. Shvydkoy  and  E. Tadmor , Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017) , 1-26.  doi: 10.1093/imatrm/tnx001.
      R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246.
      E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130401, 22pp. doi: 10.1098/rsta.2013.0401.
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