Well-posedness for systems of self-propelled particles

Abstract

This paper deals with the existence and uniqueness of solutions to kinetic equations describing alignment of self-propelled particles. The particularity of these models is that the velocity variable is not on the Euclidean space but constrained on the unit sphere (the self-propulsion constraint). Two related equations are considered: the first one, in which the alignment mechanism is nonlocal, using an observation kernel depending on the space variable, and a second form, which is purely local, corresponding to the principal order of a scaling limit of the first one. We prove local existence and uniqueness of weak solutions in both cases for bounded initial conditions (in space and velocity) with finite total mass. The solution is proven to depend continuously on the initial data in $ L^p $ spaces with finite $ p $. In the case of a bounded kernel of observation, we obtain that the solution is global in time. Finally, by exploiting the fact that the second equation corresponds to the principal order of a scaling limit of the first one, we deduce a Cauchy theory for an approximate problem approaching the second one.