Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics

Abstract

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 as $t\to \infty$.

Fix a small gap in the proof of Proposition 2.15 leading to Case 2 of this proof

References in corpus (2)

• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions

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