Well-posedness of aggregation-diffusion systems with irregular kernels

TL;DR

This paper establishes the well-posedness, including existence, uniqueness, and regularity, of solutions to aggregation-diffusion equations with irregular, bounded nonlocal kernels, using novel approaches that handle non-differentiability and non-positivity.

Contribution

It introduces new methods to prove well-posedness for aggregation-diffusion systems with irregular kernels, including cases without smallness assumptions and for multi-species systems.

Findings

Existence of weak solutions under small mass or symmetric BV kernels.

Solutions are smooth and unique when $ abla K * K$ is in $L^2$.

Constructed classical solutions for characteristic function kernels.

Abstract

We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential $K$. We are interested in establishing their well-posedness theory when the nonlocal interaction potential $K$ is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that $\nabla K\ast K$ is in $L^2$, we can prove that the solution is smooth and unique. When $K$ is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the $n$-species system.