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Eulerian dynamics with a commutator forcing Ⅱ: Flocking
1. | Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois, Chicago, IL 60607, USA |
2. | Center for Scientific Computation and Mathematical Modeling (CSCAMM), Department of Mathematics, Institute for Physical Sciences and Technology, University of Maryland, College Park, MD 20742-4015, USA |
3. | Current address: Institute for Theoretical Studies (ITS), ETH-Zurich, Clausiusstrasse 47, CH-8092 Zurich, Switzerland |
We continue our study of one-dimensional class of Euler equations, introduced in [
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 $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.
References:
[1] |
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. |
[2] |
J. Carrillo, Y.-P. Choi and S. Perez,
A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles, Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET), (2017), 259-298.
doi: 10.1007/978-3-319-49996-3_7. |
[3] |
P. Constantin and V. Vicol,
Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.
doi: 10.1007/s00039-012-0172-9. |
[4] |
T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155. Google Scholar |
[5] |
S.-Y. Ha and E. Tadmor,
From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
[6] |
C. Imbert, R. Shvydkoy and F. Vigneron,
Global well-posedness of a non-local Burgers equation: The periodic case, Annales mathématiques de Toulouse, 25 (2016), 723-758.
doi: 10.5802/afst.1509. |
[7] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2{D} dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[8] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[9] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[10] |
R.W. Schwab and L. Silvestre,
Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, 9 (2016), 727-772.
doi: 10.2140/apde.2016.9.727. |
[11] |
R. Shvydkoy and E. Tadmor,
Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017), 1-26.
doi: 10.1093/imatrm/tnx001. |
[12] |
R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246. Google Scholar |
[13] |
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. |
show all references
References:
[1] |
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. |
[2] |
J. Carrillo, Y.-P. Choi and S. Perez,
A review on attractive-repulsive hydrodynamics for consensus in collective behavior, Active Particles, Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET), (2017), 259-298.
doi: 10.1007/978-3-319-49996-3_7. |
[3] |
P. Constantin and V. Vicol,
Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.
doi: 10.1007/s00039-012-0172-9. |
[4] |
T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, arXiv: 1701.05155. Google Scholar |
[5] |
S.-Y. Ha and E. Tadmor,
From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
[6] |
C. Imbert, R. Shvydkoy and F. Vigneron,
Global well-posedness of a non-local Burgers equation: The periodic case, Annales mathématiques de Toulouse, 25 (2016), 723-758.
doi: 10.5802/afst.1509. |
[7] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2{D} dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[8] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Stat. Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[9] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[10] |
R.W. Schwab and L. Silvestre,
Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, 9 (2016), 727-772.
doi: 10.2140/apde.2016.9.727. |
[11] |
R. Shvydkoy and E. Tadmor,
Eulerian dynamics with a commutator forcing, Trans. Math. and Appl., (2017), 1-26.
doi: 10.1093/imatrm/tnx001. |
[12] |
R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing Ⅲ: Fractional diffusion of order 0 < α < 1, arXiv: 1706.08246. Google Scholar |
[13] |
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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