聚集扩散模型的光滑效应与唯一性
Smoothing effect and uniqueness for aggregation diffusion models
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中文总结 AI 辅助
针对$\mathbb R^d$（$d\geq2$）中含牛顿或贝塞尔势的聚集扩散模型，通过JKO格式分析，建立$L^\infty$光滑效应，证明梯度流解的唯一性，刻画梯度流并得到公平竞争下解的一致消失性。
中文摘要 AI 辅助
我们考虑$\mathbb R^d$（$d\geq2$）中质量密度的聚集扩散模型，其中扩散可以是线性或多孔介质型，而聚集效应由吸引性牛顿势或贝塞尔势控制。众所周知，该动力学可解释为系统自然自由能的Wasserstein梯度流，且在扩散指数和初始数据质量的适当假设下，解全局存在。在这些情形中，我们通过JKO格式对离散变分方法进行分析，建立了尖锐的$L^\infty$光滑效应，其速率与多孔介质方程的光滑效应速率相同。借助该估计，我们得到了有限能量初始数据的全局梯度流解的唯一性，且在扩散主导的情形下，得到了具有有限二阶矩的测度数据的全局梯度流解的唯一性。我们证明了能量耗散等式，并通过合适的演化变分不等式刻画了梯度流。在公平竞争情形下，我们证明了在质量的小性条件下，解随时间一致消失。
英文摘要
We consider aggregation-diffusion models for a density of mass in $\mathbb R^d$, $d\ge 2$, where the diffusion can be either linear or of porous medium type, while the aggregation effect is governed by the attractive Newtonian or Bessel potential. It is well known that the dynamics can be interpreted as Wasserstein gradient flow of the natural free energy of the system, and that, under suitable assumptions on the diffusion exponent and the mass of the initial datum, solutions exist globally in time. In these regimes, we perform the analysis of the discrete variational approach by means of the JKO scheme. We establish a sharp $L^\infty$ smoothing effect, which proves to be of same rate as that of the porous medium equation. Thanks to this estimate, we obtain uniqueness of global gradient flow solutions for initial data of finite energy and, in the diffusion dominated regime, for measure data having finite second moment. We prove the energy dissipation equality and characterize the gradient flow in terms of suitable evolution variational inequalities. In the fair competition regime, we show uniform extinction of solutions for large time under smallness conditions on the mass.
