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 non-increasing 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 ∞ 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.

中文翻译：

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扩散主导体制中的基态。

我们考虑对粒子的宏观描述，其中斥力是通过非线性幂律扩散和均质奇异核吸引来建模的，从而导致趋化性Keller-Segel模型的变体。我们分析了扩散力比粒子之间的吸引力强的状态（称为扩散主导状态），并表明系统的所有静止状态都是径向对称的非增加且紧凑的。该模型可以公式化为自由能函数的梯度流，对于该函数，其总体凸度特性未知。我们表明，全球最小的自由能一直存在。此外，它们是径向对称的，紧密支撑的，有界的并且其支撑内部的C∞。如果扩散不是太慢，则全局极小值将具有某些规则性，并且在这种情况下，将提供系统的稳定状态。在一个维度上，稳态被描述为功能不等式的优化子，它在全局极小值和稳态之间建立了等价关系，并可以推论出唯一性。