On a Class of Diffusion-Aggregation Equations

TL;DR

This paper studies diffusion-aggregation equations with degenerate diffusion and singular kernels, establishing existence, boundedness, regularity, and uniqueness of solutions in specific parameter regimes.

Contribution

It provides new results on the existence, regularity, and uniqueness of solutions for a class of diffusion-aggregation equations with degenerate diffusion and singular kernels.

Findings

Existence of solutions in certain regimes

Uniform boundedness and Hölder regularity of solutions

Uniqueness for kernels with s>1

Abstract

We investigate the diffusion-aggregation equations with degenerate diffusion $\Delta u^m$ and singular interaction kernel $\mathcal{K}_s = (-\Delta)^{-s}$ with $s\in(0,\frac{d}{2})$. We analyze the regime %($m>2-2s/d$, $d$ is the dimension) where the diffusive forces are stronger than the aggregation forces. In such regime, we show existence, uniform boundedness and H\"{o}lder regularity of solutions in the case that either $s>\frac{1}{2}$ or $m<2$. Uniqueness is proved for kernels with $s>1$.