Local Existence of Weak Solutions to Kinetic Models of Granular Media

We prove in any dimension a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media, partial derivative(t) f + v. del(x) f = div(v) [f (del W *(v) f)] when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential is a radially symmetric convex function. Our proof is constructive and relies on a splitting argument in position and velocity, where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic Wasserstein distance. Our result generalizes the local existence result obtained by Benedetto et al. (RAIRO Mod,l Math Anal Num,r 31(5):615-641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential.