A splitting method for nonlinear diffusions with nonlocal, nonpotential drifts

TL;DR

This paper introduces a novel splitting method for solving nonlinear diffusion equations with nonlocal, nonpotential drifts, combining transport and minimization steps to ensure existence of solutions.

Contribution

It presents a constructive proof of existence using a Helmholtz decomposition-based splitting scheme that handles nonlocal, nonpotential drifts in nonlinear diffusion equations.

Findings

Proves existence of solutions for complex nonlinear diffusions

Develops a splitting scheme combining transport and minimization

Handles nonlocal, nonpotential drift terms effectively

Abstract

We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density-dependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps \`a la Jordan-Kinderlherer-Otto to deal with the potential part.