  We first prove the well-posedness of weak and strong solutions to the kinetic
Cucker-Smale model in the Sobolev space with the initial data having compact
velocity support. Then we study the kinetic Cucker-Smale model with noise. By
introducing two weighted Hilbert spaces, we successfully establish the
well-posedness of weak and strong solutions, respectively. Our proof is based
on weighted energy estimates. Finally, we rigorously justify the vanishing
noise limit.
