Generalized solutions of a kinetic granular media equation by a gradient flow approach

TL;DR

This paper develops a gradient flow framework for a one-dimensional kinetic granular media model with quadratic interactions, establishing global well-posedness for measure solutions through a Wasserstein gradient flow structure.

Contribution

It introduces a reformulation leveraging a first integral to handle measure solutions and proves global existence using a Wasserstein gradient flow approach.

Findings

Global well-posedness for measure solutions

Reformulation based on a first integral

Gradient flow structure in Wasserstein space

Abstract

We consider a one-dimensional kinetic model of granular media in the case where the interaction potential is quadratic. Taking advan- tage of a simple first integral, we can use a reformulation (equivalent to the initial kinetic model for classical solutions) which allows mea- sure solutions. This reformulation has a Wasserstein gradient flow structure (on a possibly infinite product of spaces of measures) for a convex energy which enables us to prove global in time well-posedness.