Uniform in time $L^{\infty}$-estimates for nonlinear aggregation-diffusion equations

TL;DR

This paper establishes uniform in time $L^{ abla}$-bounds for solutions to nonlinear aggregation-diffusion equations with various potential singularities, using Sobolev inequalities and fractional operator techniques.

Contribution

It provides the first unified approach to derive uniform bounds for solutions under different dominance regimes of attraction, repulsion, and diffusion.

Findings

Solutions have uniform in time $L^{ abla}$-bounds under certain potential conditions.

Classical Sobolev and Young inequalities are effective for weaker singularities.

Fractional operator properties are utilized for stronger singularities.

Abstract

We derive uniform in time $L^\infty$-bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the attractive part, or the diffusion term dominates over the fully attractive nonlocal part. When the attractive potential has a weaker singularity ($2-n\leq B<A\leq2$), we use the classical approach by the Sobolev and Young inequalities together with differential iterative inequalities to prove that solutions have the uniform in time $L^{\infty}$-bound. When the repulsive potential has a stronger singularity ($-n<B<2-n\leq A\leq 2$), we show the uniform bounds by utilizing properties of fractional operators.