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Displaying items by tag: flocking

 

Trevor M. Leslie, and Changhui Tan

Communications in Partial Differential Equations, Volume 48, No. 5, pp. 753-791 (2023)


Abstract

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density and bounded velocity. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker-Smale dynamics. Our approach is largely inspired by the work of Brenier and Grenier [SIAM J. Numer. Anal, 35(6):2317-2328, 1998] on the pressureless Euler equations.


   doi:10.1080/03605302.2023.2202720
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 This work is supported by NSF grant DMS #1853001 and DMS #2108264
Published in Research

 

Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan

Archive for Rational Mechanics and Analysis, Volume 228, No 1, pp. 1-37 (2018).


Abstract

We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian \((-\partial_{xx})^{\alpha/2}, \alpha\in(0,1)\). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all \(\alpha\in(0,1)\). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.


   doi:10.1007/s00205-017-1184-2
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Published in Research