Calculus of Variations and Geometric Measure Theory
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J. A. Carrillo - F. Hoffmann - E. Mainini - B. Volzone

Ground States in the Diffusion-Dominated Regime

created by mainini on 26 May 2017
modified on 19 Jul 2018


Accepted Paper

Inserted: 26 may 2017
Last Updated: 19 jul 2018

Journal: Calc. Var. Partial Differential Equations
Year: 2018

ArXiv: 1705.03519 PDF


We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric decreasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and $C^\infty$ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

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