Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

• 成果类型: Article
• 署名作者: Carrillo, J. A.; Hittmeir, S.; Volzone, B.; Yao, Y.
• 署名单位: Imperial College London; University of Vienna; Parthenope University Naples; University System of Georgia; Georgia Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00898-x
• 发表日期: 2019
• 页码: 889-977
• 关键词: keller-segel model; preventing blow-up; interacting particles; global existence; nonlocal model; CRITICAL MASS; symmetrization; SYSTEM; chemotaxis; CONVERGENCE

摘要

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations