Nonexistence of the Asymptotic Flocking in the Cucker$-$Smale Model With Short Range Communication Weights

Abstract

For the long range communicated Cucker–Smale model, the asymptotic flocking exists for any initialcondition. It is noted that, for the short range communicated Cucker–Smale model, the asymptotic flocking only holds for very restricted initial conditions. In this case, the nonexistence of the asymptotic flocking has been frequently observed in numerical simulations, however, the theoretical results are far from perfect. In this note, we first point out that the nonexistence of the asymptotic flocking is equivalent to the unboundedness of the second order space moment, i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sup _t\sum |x_i(t)-x_j(t)|^2=\infty$</tex-math></inline-formula> . Furthermore, by taking the second derivative and then integrating, we establish a new and key equality about this moment. At last, we use this equality and relevant technical lemmas to deduce a general sufficient condition to the nonexistence of the asymptotic flocking.