Displaying items by tag: Singular interaction
On the global classical solution to compressible Euler system with singular velocity alignment
Li Chen, Changhui Tan, and Lining Tong
Methods and Applications of Analysis, Volume 28, No.2, pp. 155-174 (2021).
Dedicated to Professor Ling Hsiao's 80th birthday.
Abstract
We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.
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doi:10.4310/MAA.2021.v28.n2.a3 |
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This work is supported by NSF grant DMS #1853001 |
On the Euler-alignment system with weakly singular communication weights
Changhui Tan
Nonlinearity, Volume 33, No 4, pp. 1907-1924 (2020).
Abstract
We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of the Cucker–Smale model on animal flocks. For the Euler-alignment system with bounded interactions, a critical threshold phenomenon is proved in Tadmor and Tan (2014 Phil. Trans. R. Soc. A 372 20130401), where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained in Do et al (2018 Arch. Ration. Mech. Anal. 228 1–37), for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system with bounded interaction. However, different global behaviors may happen for critical initial data, which reveals the unique structure of the weakly singular alignment operator.
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doi:10.1088/1361-6544/ab6c39 |
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This work is supported by NSF grant DMS #1853001 |
Global regularity for 1D Eulerian dynamics with singular interaction forces
Alexander Kiselev, and Changhui Tan
SIAM Journal on Mathematical Analysis, Volume 50, No 6, pp. 6208–6229 (2018).
Abstract
The Euler–Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attrac- tion/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in [Do et al., Arch. Ration. Mech. Anal., 228 (2018), pp. 1–37]. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.
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doi:10.1137/17M1141515 |
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This work is supported by NSF grant DMS #1853001 |