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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional TCS with a singular kernel, which can be reduced to a sufficient framework for the uniform stability of the <inline-formula><tex-math id="M3">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional CS with singular kernel if all agents have the same initial temperature.</p>