Non-existence of the asymptotic flocking in the Cucker-Smale model with short range communication weights

2021 
For the long range communicated Cucker $-$ Smale model, asymptotic flocking exists for any initial data. It is noted that, for the short range communicated Cucker-Smale model, asymptotic flocking only holds for very restricted initial data. In this case, the non-existence 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 non-existence of the asymptotic flocking is equivalent to the unboundedness of the second order space moment, i.e. $\sup_t\sum|x_i(t)-x_j(t)|^2=\infty$ . 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 some technical lemmas to deduce a general sufficient condition to the non-existence of asymptotic flocking.
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