论文标题
Patlak-keller-segel模型及其固定状态的不可压缩极限
Incompressible limits of Patlak-Keller-Segel model and its stationary state
论文作者
论文摘要
我们完成了有关$ n $维$（n \ geq3）$可压缩patlak-keller-segel（PKS）型号及其固定状态的不可压缩限制的先前结果。与以前的工作一样，在此限制下，我们得出了Hele-Shaw类型的几何自由边界问题的弱形式，也称为拥挤流。特别是，我们能够考虑到不饱和区域，并建立互补关系，该关系描述了通过退化椭圆方程来描述极限压力。我们的分析不仅使用了与以前的方法完全不同的框架，而且我们还建立了压力梯度的新颖均匀$ l^3 $估计，规律性 - la aronson-bénilan和均匀的$ l^1 $估计在压力的时间导数中。此外，对于Hele-Shaw问题，我们证明了解决方案的独特性，这意味着PKS模型的不可压缩极限是唯一的。此外，我们建立了具有给定质量的PKS模型的固定状态的相应不可压缩的极限，其中与PKS模型的情况不同，我们获得了压力的均匀结合和密度均匀边界的支撑。
We complete previous results about the incompressible limit of both the $n$-dimensional $(n\geq3)$ compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish a novel uniform $L^3$ estimate of the pressure gradient, regularity à la Aronson-Bénilan, and a uniform $L^1$ estimate for the time derivative of the pressure. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.
