  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.

Analysis of PDEs