Local Existence of Weak Solutions to Kinetic Models of Granular Media

Abstract

We prove in any dimension \({d \geqq 1}\) a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media, $$\partial_t f+v\cdot \nabla_x f = {div}_v[f(\nabla W *_v f)]$$ when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential \({W}\) is a \({C^2({\mathbb{R}}^d)}\) 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 Model Math Anal Numer 31(5):615–641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential.