A new proof of uniqueness of solutions to kinetic Cucker–Smale model based on optimal transport methods

Abstract

In this paper, we will show how to extend the method used in Loeper (2006) to prove the uniqueness of weak solutions to kinetic Cucker–Smale model based on optimal transportation. Carrillo and Rosado (2010) and Loeper (2006) results show that nonnegative mass-preserving solutions remains unique as long as their L∞-norm is controlled in time. Our proof does not depend on the boundedness of density, only on the conservation of mass and the boundedness of the second moment of velocity. Compared with the Carrillo and Rosado (2010) result, we study the potential function ∇K∉L2(R2d).