Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels

Hyunjin Ahn; Hyunjin Ahn

doi:10.3934/nhm.2022025

Networks and Heterogeneous Media

2022, Volume 17, Issue 5: 753-782. doi: 10.3934/nhm.2022025

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Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels

Hyunjin Ahn ^,

Department of Mathematical Sciences, and Research Institute of Mathematics, Seoul National University, Seoul, Republic of Korea

Received: 01 March 2022 Revised: 01 May 2022 Published: 16 June 2022
34D23, 34E99, 82C22

This paper presents several sufficient frameworks for a collision avoidance and flocking dynamics of the Cucker–Smale (CS) model and thermodynamic CS (TCS) model with arbitrary dimensions and singular interaction kernels. In general, unlike regular kernels, singular kernels usually interfere with the global well-posedness of the targeted models from the perspective of the standard Cauchy–Lipschitz theory due to the possibility of a finite-in-time blow-up. Therefore, according to the intensity of the singularity of a kernel (strong or weak), we provide a detailed framework for the global well-posedness and emergent dynamics for each case. Finally, we provide an admissible set in terms of system parameters and initial data for the uniform stability of the $ d $-dimensional TCS with a singular kernel, which can be reduced to a sufficient framework for the uniform stability of the $ d $-dimensional CS with singular kernel if all agents have the same initial temperature.
- Asymptotic flocking,
- collision avoidance,
- Cucker–Smale,
- strongly singular kernel,
- thermodynamic Cucker–Smale,
- uniform $ L^2 $-stability,
- weakly singular kernel
Citation: Hyunjin Ahn. Uniform stability of the Cucker–Smale and thermodynamic Cucker–Smale ensembles with singular kernels[J]. Networks and Heterogeneous Media, 2022, 17(5): 753-782. doi: 10.3934/nhm.2022025

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Abstract

This paper presents several sufficient frameworks for a collision avoidance and flocking dynamics of the Cucker–Smale (CS) model and thermodynamic CS (TCS) model with arbitrary dimensions and singular interaction kernels. In general, unlike regular kernels, singular kernels usually interfere with the global well-posedness of the targeted models from the perspective of the standard Cauchy–Lipschitz theory due to the possibility of a finite-in-time blow-up. Therefore, according to the intensity of the singularity of a kernel (strong or weak), we provide a detailed framework for the global well-posedness and emergent dynamics for each case. Finally, we provide an admissible set in terms of system parameters and initial data for the uniform stability of the $ d $-dimensional TCS with a singular kernel, which can be reduced to a sufficient framework for the uniform stability of the $ d $-dimensional CS with singular kernel if all agents have the same initial temperature.

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