The flocking behavior of the infinite-particle Cucker-Smale model
Xinyu Wang, Xiaoping Xue
Abstract
In this article, we study the flocking behavior of the solutions to the infinite-particle Cucker-Smale model. We first establish the existence and uniqueness of the solutions to the infinite-particle Cucker-Smale model. And then build the boundedness of velocity by showing the non-increase of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>l</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">l_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">v(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> through classifying the particles according to the norm of velocity. Finally, we obtain the flocking behavior of the infinite-particle Cucker-Smale model. More precisely, the solutions to the infinite-particle Cucker-Smale model will concentrate exponentially fast in velocity to the average of the initial velocity, while in space the position differences between particles will be uniformly bounded.