Uniform in Time L ∞ $L^{\infty }$ -Estimates for Nonlinear Aggregation-Diffusion Equations

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 repulsive part of the potential has a weaker singularity ( $2-n\leq B< A\leq 2$ ), 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 part of the potential has a stronger singularity ( $-n< B<2-n \leq A\leq 2$ ), we show the uniform bounds by utilizing properties of fractional operators. We also show uniform bounds in the purely attractive case $2-n\leq A\leq 2$ within the diffusion dominated regime.