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

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arxiv 1603.07767 v3 pith:OUJYOZ5V submitted 2016-03-24 math.AP

J. A. Carrillo , S. Hittmeir , B. Volzone , Y. Yao This is my paper · ORCID
classification math.AP
keywords nonlinearaggregation-diffusioncontinuousequationsequilibriaequilibriuminteractionmodeled
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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 $t\to\infty$.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations

    math.AP 2019-08 accept novelty 8.0 of 10

    For aggregation-diffusion equations with attractive potentials, this paper proves steady states of fixed mass are unique iff m >= 2, and constructs non-uniqueness examples for 1 < m < 2.

  2. Symmetry in stationary and uniformly-rotating solutions of active scalar equations

    math.AP 2019-08 accept novelty 8.0 of 10

    Any uniformly rotating 2D Euler patch with angular velocity at most 0 or at least 1/2, and any simply connected gSQG patch with sufficiently large angular velocity, must be radially symmetric.

  3. Equilibria of an aggregation model with linear diffusion in domains with boundaries

    math.AP 2019-08 conditional novelty 7.0 of 10

    Aggregation-diffusion energies on domains with boundaries have an existence threshold set by the effective volume dimension, and asymmetric unbounded domains admit no minimizer without external confinement.

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