Uniform in time 𝐿∞-estimates for nonlinear aggregation-diffusion equations

Journal article

We derive uniform in time 𝐿∞-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≤𝐵<𝐴≤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∞-bound. When the repulsive part of the potential has a stronger singularity (−𝑛<𝐵<2−𝑛≤𝐴≤2), we show the uniform bounds by utilizing properties of fractional operators. We also show uniform bounds in the purely attractive case 2−𝑛≤𝐴≤2 within the diffusion dominated regime.

Authors: Carrillo, JA; Wang, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Acta Applicandae Mathematicae
• Volume: 164
• Issue: 1
• Pages: 1-19
• Publication date: 2018-10-26
• Acceptance date: 2018-10-21
• DOI: 10.1007/s10440-018-0221-y
• EISSN: 1572-9036
• ISSN: 0167-8019
• Language: English
• Keywords: FFR; aggregation-diffusion equations; Stroock-Varopoulos inequality; global in time uniform estimates