On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility

TL;DR

This paper establishes the well-posedness of entropy solutions for nonlocal transport equations with nonlinear mobility in one dimension, using particle approximations and revealing a gradient flow structure.

Contribution

It introduces a novel approach combining particle systems and gradient flow analysis to study these complex equations.

Findings

Proves well-posedness of entropy solutions.

Shows convergence of particle approximations.

Identifies a gradient flow structure via Energy-Dissipation balance.

Abstract

We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. The solution is obtained as the limit of approximations constructed via a deterministic system of interacting particles that exhibits a gradient flow structure. At the same time, we expose a rigorous gradient flow structure for this class of equations in terms of an Energy-Dissipation balance, which we obtain via the asymptotic convergence of functionals.